public static MathObject InitialAngle(Obj a, Obj b, int solution = 0, int n = 0)
{
if (Misc.NotNull(
a.position.x,
b.position.x,
a.position.y,
b.position.y,
a.speed,
a.acceleration.y)
&&
a.speed != 0 &&
a.speed != 0.0)
{
var xf = b.position.x - a.position.x;
var yf = b.position.y - a.position.y;
var vi = a.speed;
var ay = a.acceleration.y;
if (solution == 0)
return
(Trig.Asin((yf - ay * (xf ^ 2) / (vi ^ 2)) / Misc.Sqrt((xf ^ 2) + (yf ^ 2))) + Trig.Atan2(yf, xf))
/
2;
else if (solution == 1)
return
(Trig.Pi - Trig.Asin((yf - ay * (xf ^ 2) / (vi ^ 2)) / Misc.Sqrt((xf ^ 2) + (yf ^ 2))) + Trig.Atan2(yf, xf))
/
2;
}
throw new Exception();
}
static void Main(string[] args)
{
// A simple accelerometer is constructed by suspending a
// mass m from a string of length L that is tied to the top
// of a cart. As the cart is accelerated the string-mass
// system makes a constant angle th with the vertical.
// (a) Assuming that the string mass is negligible compared
// with m, derive an expression for the cart’s acceleration
// in terms of and show that it is independent of
// the mass mand the length L.
// (b) Determine the acceleration of the cart when th = 23.0°.
var F1 = new Symbol("F1"); // force of string
var F2 = new Symbol("F2"); // force of gravity
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");;
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2, magnitude = F2 };
var m = new Symbol("m");
var g = new Symbol("g");
var obj = new Obj() { mass = m };
obj.acceleration.y = 0;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
_F1.magnitude = obj.ForceMagnitude(_F1);
("Derive an expression for the cart’s acceleration in terms " +
"of and show that it is independent of the mass mand the length L:").Disp();
"".Disp();
obj.AccelerationX()
.Substitute(F2, m * g)
.Substitute(Trig.Cos(th2), 0)
.Substitute(Trig.Sin(th2), -1)
.Disp();
"".Disp();
"Determine the acceleration of the cart when th = 23.0°".Disp(); "".Disp();
obj.AccelerationX()
.Substitute(F2, m * g)
.Substitute(Trig.Cos(th2), 0)
.Substitute(Trig.Sin(th2), -1)
.Substitute(th1, (90 - 23).ToRadians())
.Substitute(Trig.Pi, Math.PI)
.Substitute(g, 9.8)
.Disp();
Console.ReadLine();
}
static void Main(string[] args)
{
// You are a judge in a children’s kite-flying contest, and
// two children will win prizes for the kites that pull most
// strongly and least strongly on their strings. To measure
// string tensions, you borrow a weight hanger, some slotted
// weights, and a protractor from your physics teacher
// and use the following protocol, illustrated in Figure
// P5.26: Wait for a child to get her kite well-controlled,
// hook the hanger onto the kite string about 30 cm from
// her hand, pile on weights until that section of string is
// horizontal, record the mass required, and record the
// angle between the horizontal and the string running up
// to the kite. (a) Explain how this method works. As you
// construct your explanation, imagine that the children’s
// parents ask you about your method, that they might
// make false assumptions about your ability without concrete
// evidence, and that your explanation is an opportunity to
// give them confidence in your evaluation tech-nique.
// (b) Find the string tension if the mass required
// to make the string horizontal is 132 g and the angle of
// the kite string is 46.3°.
var F1 = new Symbol("F1");
var F2 = new Symbol("F2");
var F3 = new Symbol("F3");
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");
var th3 = new Symbol("th3");
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2 };
var _F3 = new Point() { magnitude = F3, angle = th3 };
var m = new Symbol("m");
var obj = new Obj();
obj.acceleration.x = 0;
obj.acceleration.y = 0;
obj.mass = m;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
obj.forces.Add(_F3);
"Tension in line to kite:".Disp(); "".Disp();
obj.ForceMagnitude(_F2)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (46.3 * Math.PI / 180))
.Substitute(th3, (270).ToRadians())
.Substitute(F3, 0.132 * 9.8)
.Substitute(Trig.Pi, Math.PI)
.Disp();
Console.ReadLine();
}
static void Main(string[] args)
{
// A cannon with a muzzle speed of 1 000 m/s is used to
// start an avalanche on a mountain slope. The target is
// 2 000 m from the cannon horizontally and 800 m above
// the cannon. At what angle, above the horizontal, should
// the cannon be fired?
var xA = new Symbol("xA"); // position.x at point A
var yA = new Symbol("yA"); // position.y at point A
var thA = new Symbol("thA"); // angle of snowball 1 at point A
var vA = new Symbol("vA"); // velocity at point A
var xB = new Symbol("xB"); // position.x at point A
var yB = new Symbol("yB"); // position.y at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA = new Obj()
{
position = new Point(xA, yA),
speed = vA,
acceleration = _g,
time = 0
};
var objB = new Obj()
{
position = new Point(xB, yB),
acceleration = _g
};
"At what angle, above the horizontal, should the cannon be fired?".Disp();
Calc.InitialAngle(objA, objB)
.ToDegrees()
.Substitute(xA, 0)
.Substitute(yA, 0)
.Substitute(xB, 2000.0)
.Substitute(yB, 800)
.Substitute(vA, 1000)
.Substitute(g, 9.8)
.Substitute(Trig.Pi, Math.PI)
.Disp();
Console.ReadLine();
}
public static MathObject ForceMagnitude(Obj a, Obj b, Point _F1A, Point _F1B)
{
if (a.forces.Count(elt => elt.magnitude == null) == 2 &&
a.forces.Count(elt => elt.angle == null) == 0 &&
b.forces.Count(elt => elt.magnitude == null) == 2 &&
b.forces.Count(elt => elt.angle == null) == 0)
{
var _F2A = a.forces.Find(elt => elt != _F1A && elt.magnitude == null);
var _F2B = b.forces.Find(elt => elt != _F1B && elt.magnitude == null);
var th1A = _F1A.angle;
var th1B = _F1B.angle;
var th2A = _F2A.angle;
var th2B = _F2B.angle;
var knownForcesA = new List<Point>(a.forces);
knownForcesA.Remove(_F1A);
knownForcesA.Remove(_F2A);
var knownForcesB = new List<Point>(b.forces);
knownForcesB.Remove(_F1B);
knownForcesB.Remove(_F2B);
var result = -b.acceleration.x * b.mass * Trig.Cos(th2A);
result += a.acceleration.x * a.mass * Trig.Cos(th2B);
knownForcesA.ForEach(_F3A =>
result -= _F3A.magnitude * Trig.Cos(_F3A.angle) * Trig.Cos(th2B));
knownForcesB.ForEach(_F3B =>
result += _F3B.magnitude * Trig.Cos(_F3B.angle) * Trig.Cos(th2A));
if ((-Trig.Cos(th1B) * Trig.Cos(th2A) + Trig.Cos(th1A) * Trig.Cos(th2B)) != 0)
{
result /= (-Trig.Cos(th1B) * Trig.Cos(th2A) + Trig.Cos(th1A) * Trig.Cos(th2B));
return result;
}
}
throw new Exception();
}
static void Main(string[] args)
{
Action<Equation> AssertIsTrue = (eq) =>
{
if (!eq) Console.WriteLine(eq.ToString());
};
Func<MathObject, MathObject> sin = obj => Trig.Sin(obj);
Func<MathObject, MathObject> cos = obj => Trig.Cos(obj);
{
var x = new Symbol("x");
var y = new Symbol("y");
var z = new Symbol("z");
Func<int, Integer> Int = (n) => new Integer(n);
AssertIsTrue(x + x == 2 * x);
AssertIsTrue(x + x == 2 * x);
AssertIsTrue(x + x + x == 3 * x);
AssertIsTrue(5 + x + 2 == 7 + x);
AssertIsTrue(3 + x + 5 + x == 8 + 2 * x);
AssertIsTrue(4 * x + 3 * x == 7 * x);
AssertIsTrue(x + y + z + x + y + z == 2 * x + 2 * y + 2 * z);
AssertIsTrue(10 - x == 10 + x * -1);
AssertIsTrue(x * y / 3 == Int(1) / 3 * x * y);
AssertIsTrue(x / y == x * (y ^ -1));
AssertIsTrue(x / 3 == x * (Int(1) / 3));
AssertIsTrue(6 * x * y / 3 == 2 * x * y);
AssertIsTrue((((x ^ Int(1) / 2) ^ Int(1) / 2) ^ 8) == (x ^ 2));
AssertIsTrue(((((x * y) ^ (Int(1) / 2)) * (z ^ 2)) ^ 2) == (x * y * (z ^ 4)));
AssertIsTrue(x / x == 1);
AssertIsTrue(x / y * y / x == 1);
AssertIsTrue((x ^ 2) * (x ^ 3) == (x ^ 5));
AssertIsTrue(x + y + x + z + 5 + z == 5 + 2 * x + y + 2 * z);
AssertIsTrue(((Int(1) / 2) * x + (Int(3) / 4) * x) == Int(5) / 4 * x);
AssertIsTrue(1.2 * x + 3 * x == 4.2 * x);
AssertIsTrue(3 * x + 1.2 * x == 4.2 * x);
AssertIsTrue(1.2 * x * 3 * y == 3.5999999999999996 * x * y);
AssertIsTrue(3 * x * 1.2 * y == 3.5999999999999996 * x * y);
AssertIsTrue(3.4 * x * 1.2 * y == 4.08 * x * y);
// Power.Simplify
AssertIsTrue((0 ^ x) == 0);
AssertIsTrue((1 ^ x) == 1);
AssertIsTrue((x ^ 0) == 1);
AssertIsTrue((x ^ 1) == x);
// Product.Simplify
AssertIsTrue(x * 0 == 0);
// Difference
AssertIsTrue(-x == -1 * x);
AssertIsTrue(x - y == x + -1 * y);
AssertIsTrue(Int(10).Substitute(Int(10), 20) == 20);
AssertIsTrue(Int(10).Substitute(Int(15), 20) == 10);
AssertIsTrue(new DoubleFloat(1.0).Substitute(new DoubleFloat(1.0), 2.0) == 2.0);
AssertIsTrue(new DoubleFloat(1.0).Substitute(new DoubleFloat(1.5), 2.0) == 1.0);
AssertIsTrue((Int(1) / 2).Substitute(Int(1) / 2, Int(3) / 4) == Int(3) / 4);
AssertIsTrue((Int(1) / 2).Substitute(Int(1) / 3, Int(3) / 4) == Int(1) / 2);
AssertIsTrue(x.Substitute(x, y) == y);
AssertIsTrue(x.Substitute(y, y) == x);
AssertIsTrue((x ^ y).Substitute(x, 10) == (10 ^ y));
AssertIsTrue((x ^ y).Substitute(y, 10) == (x ^ 10));
AssertIsTrue((x ^ y).Substitute(x ^ y, 10) == 10);
AssertIsTrue((x * y * z).Substitute(x, y) == ((y ^ 2) * z));
AssertIsTrue((x * y * z).Substitute(x * y * z, x) == x);
AssertIsTrue((x + y + z).Substitute(x, y) == ((y * 2) + z));
AssertIsTrue((x + y + z).Substitute(x + y + z, x) == x);
AssertIsTrue(
((((x * y) ^ (Int(1) / 2)) * (z ^ 2)) ^ 2)
.Substitute(x, 10)
.Substitute(y, 20)
.Substitute(z, 3)
== 16200
);
AssertIsTrue(sin(new DoubleFloat(3.14159 / 2)) == 0.99999999999911982);
AssertIsTrue(sin(x + y) + sin(x + y) == 2 * sin(x + y));
AssertIsTrue(sin(x + x) == sin(2 * x));
AssertIsTrue(sin(x + x).Substitute(x, 1) == sin(Int(2)));
AssertIsTrue(sin(x + x).Substitute(x, 1.0) == 0.90929742682568171);
AssertIsTrue(sin(2 * x).Substitute(x, y) == sin(2 * y));
// Product.RecursiveSimplify
AssertIsTrue(1 * x == x);
AssertIsTrue(x * 1 == x);
AssertIsTrue(x != y);
AssertIsTrue(x != 10);
// ==(double a, MathObject b)
AssertIsTrue(1.0 == new DoubleFloat(3.0) - 2.0);
// Console.WriteLine((x + x + x + x) / x);
}
#region PSE 5E Example 4.3
{
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA =
new Obj()
{
position = new Point(0, 0),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = new Integer(0)
};
var objB =
new Obj()
{
velocity = new Point(objA.velocity.x, 0),
acceleration = _g
};
var timeB = Calc.Time(objA, objB);
var timeC = timeB * 2;
objB = objA.AtTime(timeB);
var objC = objA.AtTime(timeC);
//Console.WriteLine("How far does he dump in the horizontal direction?");
AssertIsTrue(objC.position.x == 2 * Trig.Cos(thA) * Trig.Sin(thA) * (vA ^ 2) / g);
//Console.WriteLine("What is the maximum height reached?");
AssertIsTrue(objB.position.y == (Trig.Sin(thA) ^ 2) * (vA ^ 2) / 2 / g);
// Console.WriteLine("Distance jumped: ");
var deg = 3.14159 / 180.0;
AssertIsTrue(
objC.position.x
// .Substitute(thA, Trig.ToRadians(20))
.Substitute(thA, 20 * deg)
.Substitute(g, 9.8)
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 11)
==
7.9364536850196412);
//Console.WriteLine("Maximum height reached: ");
AssertIsTrue(
objB.position.y
.Substitute(g, 9.8)
.Substitute(thA, Trig.ToRadians(20))
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 11)
==
0.72215756424454336);
}
#endregion
#region PSE 5E EXAMPLE 4.5
{
// A stone is thrown from the top of a building upward at an
// angle of 30.0° to the horizontal and with an initial speed of
// 20.0 m/s, as shown in Figure 4.12. If the height of the building
// is 45.0 m, (a) how long is it before the stone hits the ground?
// (b) What is the speed of the stone just before it strikes the
// ground?
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA = new Obj()
{
position = new Point(0, 0),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = new Integer(0)
};
var objB = new Obj()
{
velocity = new Point(objA.velocity.x, 0),
acceleration = _g,
};
var timeB = Calc.Time(objA, objB);
objB = objA.AtTime(timeB);
var timeC = timeB * 2;
var objC = objA.AtTime(timeC);
var yD = new Symbol("yD");
var objD = new Obj()
{
position = new Point(null, yD),
velocity = new Point(objA.velocity.x, null),
acceleration = _g
};
var timeAD = Calc.Time(objA, objD, 1);
objD = objA.AtTime(timeAD);
// "How long is it before the stone hits the ground?".Disp();
// "Symbolic answer:".Disp();
AssertIsTrue(
timeAD
==
-1 * (g ^ -1) * (-1 * Trig.Sin(thA) * vA + -1 * (((Trig.Sin(thA) ^ 2) * (vA ^ 2) + -2 * g * yD) ^ (new Integer(1) / 2))));
// "Numeric answer:".Disp();
AssertEqual(
(DoubleFloat)
timeAD
.Substitute(g, 9.8)
.Substitute(thA, (30).ToRadians())
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 20)
.Substitute(yD, -45),
new DoubleFloat(4.21804787012706),
0.0001);
// "What is the speed of the stone just before it strikes the ground?".Disp();
// "Symbolic answer:".Disp();
AssertIsTrue(
objD.velocity.Norm()
==
(((Trig.Cos(thA) ^ 2) * (vA ^ 2) + (Trig.Sin(thA) ^ 2) * (vA ^ 2) + -2 * g * yD) ^ (new Integer(1) / 2)));
// "Numeric answer:".Disp();
AssertEqual(
(DoubleFloat)
objD.velocity.Norm()
.Substitute(g, 9.8)
.Substitute(thA, (30).ToRadians())
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 20)
.Substitute(yD, -45),
new DoubleFloat(35.805027579936315),
0.1);
}
#endregion
#region PSE 5E EXAMPLE 4.6
{
// An Alaskan rescue plane drops a package of emergency rations
// to a stranded party of explorers, as shown in Figure
// 4.13. If the plane is traveling horizontally at 40.0 m/s and is
// 100 m above the ground, where does the package strike the
// ground relative to the point at which it was released?
var xA = new Symbol("xA"); // position.x at point A
var yA = new Symbol("yA"); // position.y at point A
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA = new Obj() // obj at the initial position
{
position = new Point(xA, yA),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = 0
};
var objB = new Obj() // obj at the final position
{
position = new Point(null, 0),
velocity = new Point(objA.velocity.x, null),
acceleration = _g
};
var timeB = Calc.Time(objA, objB, 1);
objB = objA.AtTime(timeB);
//"Where does the package strike the ground relative to the point at which it was released?".Disp(); "".Disp();
//"symbolic:".Disp();
//objB.position.x.Disp(); "".Disp();
AssertIsTrue(
objB.position.x
==
xA - cos(thA) / g * vA * (-sin(thA) * vA - (((sin(thA) ^ 2) * (vA ^ 2) + 2 * g * yA) ^ new Integer(1) / 2)));
//"numeric:".Disp();
//objB.position.x
// .Substitute(xA, 0)
// .Substitute(yA, 100)
// .Substitute(vA, 40)
// .Substitute(thA, 0.0)
// .Substitute(g, 9.8)
// .Disp();
AssertEqual(
objB.position.x
.Substitute(xA, 0)
.Substitute(yA, 100)
.Substitute(vA, 40)
.Substitute(thA, 0.0)
.Substitute(g, 9.8),
180.70158058105025);
//"".Disp();
//("What are the horizontal and vertical components " +
// "of the velocity of the package just before it hits the ground?").Disp(); "".Disp();
//"symbolic velocity.x:".Disp();
//objB.velocity.x.Disp(); "".Disp();
AssertIsTrue(objB.velocity.x == cos(thA) * vA);
//"symbolic velocity.y:".Disp();
//objB.velocity.y.Disp(); "".Disp();
AssertIsTrue(
objB.velocity.y
==
-1 * (((sin(thA) ^ 2) * (vA ^ 2) + 2 * g * yA) ^ (new Integer(1) / 2)));
//"numeric velocity.x:".Disp();
//objB.velocity.x
// .Substitute(xA, 0)
// .Substitute(yA, 100)
// .Substitute(vA, 40)
// .Substitute(thA, 0.0)
// .Substitute(g, 9.8)
// .Disp(); "".Disp();
AssertEqual(
objB.velocity.x
.Substitute(xA, 0)
.Substitute(yA, 100)
.Substitute(vA, 40)
.Substitute(thA, 0.0)
.Substitute(g, 9.8),
40);
//"numeric velocity.y:".Disp();
//objB.velocity.y
// .Substitute(xA, 0)
// .Substitute(yA, 100)
// .Substitute(vA, 40)
// .Substitute(thA, 0.0)
// .Substitute(g, 9.8)
// .Disp(); "".Disp();
AssertEqual(
objB.velocity.y
.Substitute(xA, 0)
.Substitute(yA, 100)
.Substitute(vA, 40)
.Substitute(thA, 0.0)
.Substitute(g, 9.8),
-44.271887242357316);
}
#endregion
#region PSE 5E EXAMPLE 4.7
{
// A ski jumper leaves the ski track moving in the horizontal
// direction with a speed of 25.0 m/s, as shown in Figure 4.14.
// The landing incline below him falls off with a slope of 35.0°.
// Where does he land on the incline?
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var th = new Symbol("th"); // angle of incline
var objA = new Obj()
{
position = new Point(0, 0),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = 0
};
Func<MathObject, MathObject> numeric = obj =>
obj
.Substitute(vA, 25)
.Substitute(thA, 0.0)
.Substitute(th, (-35).ToRadians())
.Substitute(Trig.Pi, 3.14159)
.Substitute(g, 9.8);
var intersection = objA.ProjectileInclineIntersection(th);
Action nl = () => "".Disp();
// "Where does he land on the incline?".Disp(); nl();
// "x position (symbolic):".Disp();
// intersection.x.Disp(); nl();
AssertIsTrue(
intersection.x
==
-2 * (cos(th) ^ -1) * (cos(thA) ^ 2) * (g ^ -1) * (sin(th) + -1 * cos(th) * (cos(thA) ^ -1) * sin(thA)) * (vA ^ 2));
//"y position (symbolic):".Disp();
//intersection.y.Disp(); nl();
AssertIsTrue(
intersection.y
==
-2 * (cos(th) ^ -2) * (cos(thA) ^ 2) / g * sin(th) * (sin(th) + -1 * cos(th) * (cos(thA) ^ -1) * sin(thA)) * (vA ^ 2));
//"x position (numeric):".Disp();
//numeric(intersection.x).Disp(); nl();
AssertEqual(numeric(intersection.x), 89.3120879153208);
//"y position (numeric):".Disp();
//numeric(intersection.y).Disp(); nl();
AssertEqual(numeric(intersection.y), -62.536928534704884);
var objB = new Obj()
{
position = intersection,
acceleration = _g
};
//"Determine how long the jumper is airborne".Disp(); nl();
//"symbolic:".Disp();
var timeB = Calc.Time(objA, objB, 1);
// timeB.Disp(); nl();
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
AssertIsTrue(
timeB
==
-1 / g *
(-sin(thA) * vA -
sqrt(
(sin(thA) ^ 2) * (vA ^ 2) + 4 * (cos(th) ^ -2) * (cos(thA) ^ 2) * sin(th) *
(sin(th) - cos(th) / cos(thA) * sin(thA)) *
(vA ^ 2))));
//"numeric:".Disp();
//numeric(timeB).Disp(); nl();
AssertEqual(numeric(timeB), 3.5724835166128317);
objB = objA.AtTime(timeB);
//"Determine his vertical component of velocity just before he lands".Disp();
//nl();
//"symbolic:".Disp();
//objB.velocity.y.Disp(); nl();
AssertIsTrue(
objB.velocity.y
==
-sqrt(
(sin(thA) ^ 2) * (vA ^ 2)
+
4 * (cos(th) ^ -2) * (cos(thA) ^ 2) * sin(th) *
(sin(th) - cos(th) * (cos(thA) ^ -1) * sin(thA)) *
(vA ^ 2)));
//"numeric:".Disp();
//numeric(objB.velocity.y).Disp();
AssertEqual(
numeric(objB.velocity.y),
-35.010338462805755);
}
#endregion
#region PSE 5E PROBLEM 4.11
{
// One strategy in a snowball fight is to throw a first snowball at a
// high angle over level ground. While your opponent is watching the
// first one, you throw a second one at a low angle and timed to arrive
// at your opponent before or at the same time as the first one. Assume
// both snowballs are thrown with a speed of 25.0 m/s. The first one is
// thrown at an angle of 70.0° with respect to the horizontal.
//
// (a) At what angle should the second (low-angle) snowball be thrown
// if it is to land at the same point as the first?
//
// (b) How many seconds later should the second snowball be thrown if it
// is to land at the same time as the first?
var xA = new Symbol("xA"); // position.x at point A
var yA = new Symbol("yA"); // position.y at point A
var th1A = new Symbol("th1A"); // angle of snowball 1 at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
//Func<MathObject, MathObject> numeric = obj =>
// obj
// .Substitute(xA, 0)
// .Substitute(xB, 1.4)
// .Substitute(yA, 0.86)
// .Substitute(g, 9.8)
// .Substitute(Trig.Pi, 3.14159);
var obj1A = new Obj() // snowball 1 at initial point
{
position = new Point(xA, yA),
velocity = Point.FromAngle(th1A, vA),
acceleration = _g,
time = 0
};
var obj1B = new Obj() // snowball 1 at final point
{
position = new Point(null, 0),
velocity = new Point(obj1A.velocity.x, null),
acceleration = _g
};
var time1B = Calc.Time(obj1A, obj1B, 1);
obj1B = obj1A.AtTime(time1B);
var obj2A = new Obj() // snowball 2 at initial point
{
position = obj1A.position,
speed = vA,
acceleration = _g
};
var obj2B = new Obj() // snowball 2 at final point
{
position = obj1B.position,
acceleration = _g
};
//Calc.InitialAngle(obj2A, obj2B, 1, 0)
// .Substitute(yA, 0)
// .Substitute(th1A, (70).ToRadians())
// .Substitute(vA, 25)
// .Substitute(Trig.Pi, 3.14159)
// .Substitute(g, 9.8)
// .ToDegrees()
// .Substitute(Trig.Pi, 3.14159)
// .Disp();
var th2 = Calc.InitialAngle(obj2A, obj2B, 0, 0);
//("At what angle should the second (low-angle) snowball " +
//"be thrown if it is to land at the same point as the first?").Disp();
//"".Disp();
//"symbolic:".Disp();
//th2.Disp(); "".Disp();
//"numeric:".Disp();
AssertEqual(
th2
.ToDegrees()
.Substitute(yA, 0)
.Substitute(th1A, (70).ToRadians())
.Substitute(vA, 25)
.Substitute(g, 9.8)
.Substitute(Trig.Pi, Math.PI),
20.000000000000007);
//"".Disp();
obj2A.velocity = Point.FromAngle(th2, vA);
var time2B = Calc.Time(obj2A, obj2B, 1);
//("How many seconds later should the second snowball be thrown if it " +
//"is to land at the same time as the first?").Disp();
//"".Disp();
//"symbolic:".Disp();
//(time1B - time2B).Disp(); "".Disp();
//"numeric:".Disp();
//(time1B - time2B)
// .Substitute(yA, 0)
// .Substitute(th1A, (70).ToRadians())
// .Substitute(vA, 25)
// .Substitute(Trig.Pi, 3.14159)
// .Substitute(g, 9.8)
// .Disp();
AssertEqual(
(time1B - time2B)
.Substitute(yA, 0)
.Substitute(th1A, (70).ToRadians())
.Substitute(vA, 25)
.Substitute(Trig.Pi, 3.14159)
.Substitute(g, 9.8),
3.0493426265020469);
//Console.ReadLine();
}
#endregion
#region PSE 5E PROBLEM 4.17
{
// A cannon with a muzzle speed of 1 000 m/s is used to
// start an avalanche on a mountain slope. The target is
// 2 000 m from the cannon horizontally and 800 m above
// the cannon. At what angle, above the horizontal, should
// the cannon be fired?
var xA = new Symbol("xA"); // position.x at point A
var yA = new Symbol("yA"); // position.y at point A
var thA = new Symbol("thA"); // angle of snowball 1 at point A
var vA = new Symbol("vA"); // velocity at point A
var xB = new Symbol("xB"); // position.x at point A
var yB = new Symbol("yB"); // position.y at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA = new Obj()
{
position = new Point(xA, yA),
speed = vA,
acceleration = _g,
time = 0
};
var objB = new Obj()
{
position = new Point(xB, yB),
acceleration = _g
};
//"At what angle, above the horizontal, should the cannon be fired?".Disp();
AssertEqual(
Calc.InitialAngle(objA, objB)
.ToDegrees()
.Substitute(xA, 0)
.Substitute(yA, 0)
.Substitute(xB, 2000.0)
.Substitute(yB, 800)
.Substitute(vA, 1000)
.Substitute(g, 9.8)
.Substitute(Trig.Pi, Math.PI),
22.365163229244317);
//Calc.InitialAngle(objA, objB)
// .ToDegrees()
// .Substitute(xA, 0)
// .Substitute(yA, 0)
// .Substitute(xB, 2000.0)
// .Substitute(yB, 800)
// .Substitute(vA, 1000)
// .Substitute(g, 9.8)
// .Substitute(Trig.Pi, Math.PI)
// .Disp();
}
#endregion
#region PSE 5E PROBLEM 4.24
{
// A bag of cement of weight 325 N hangs from three
// wires as shown in Figure P5.24. Two of the wires make
// angles th1 = 60.0° and th2 = 25.0° with the horizontal. If
// the system is in equilibrium, find the tensions
// T1, T2, and T3 in the wires.
var F1 = new Symbol("F1");
var F2 = new Symbol("F2");
var F3 = new Symbol("F3");
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");
var th3 = new Symbol("th3");
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2 };
var _F3 = new Point() { magnitude = F3, angle = th3 };
var m = new Symbol("m");
var obj = new Obj();
obj.acceleration.x = 0;
obj.acceleration.y = 0;
obj.mass = m;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
obj.forces.Add(_F3);
//"F1 magnitude, symbolic:".Disp(); "".Disp();
//obj.ForceMagnitude(_F1).Disp(); "".Disp();
//"F1 magnitude, numeric:".Disp(); "".Disp();
//obj.ForceMagnitude(_F1)
// .Substitute(F3, 325)
// .Substitute(th1, (180 - 60).ToRadians())
// .Substitute(th2, (25).ToRadians())
// .Substitute(th3, (270).ToRadians())
// .Substitute(Trig.Pi, Math.PI)
// .Disp();
AssertEqual(
obj.ForceMagnitude(_F1)
.Substitute(F3, 325)
.Substitute(th1, (180 - 60).ToRadians())
.Substitute(th2, (25).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI),
295.67516405290525);
// "".Disp();
//"F3 magnitude, numeric:".Disp(); "".Disp();
//obj.ForceMagnitude(_F2)
// .Substitute(F3, 325)
// .Substitute(th1, (180 - 60).ToRadians())
// .Substitute(th2, (25).ToRadians())
// .Substitute(th3, (270).ToRadians())
// .Substitute(Trig.Pi, Math.PI)
// .Disp();
AssertEqual(
obj.ForceMagnitude(_F2)
.Substitute(F3, 325)
.Substitute(th1, (180 - 60).ToRadians())
.Substitute(th2, (25).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI),
163.12072360079395);
}
#endregion
#region PSE 5E PROBLEM 5.26
{
// You are a judge in a children’s kite-flying contest, and
// two children will win prizes for the kites that pull most
// strongly and least strongly on their strings. To measure
// string tensions, you borrow a weight hanger, some slotted
// weights, and a protractor from your physics teacher
// and use the following protocol, illustrated in Figure
// P5.26: Wait for a child to get her kite well-controlled,
// hook the hanger onto the kite string about 30 cm from
// her hand, pile on weights until that section of string is
// horizontal, record the mass required, and record the
// angle between the horizontal and the string running up
// to the kite. (a) Explain how this method works. As you
// construct your explanation, imagine that the children’s
// parents ask you about your method, that they might
// make false assumptions about your ability without concrete
// evidence, and that your explanation is an opportunity to
// give them confidence in your evaluation tech-nique.
// (b) Find the string tension if the mass required
// to make the string horizontal is 132 g and the angle of
// the kite string is 46.3°.
var F1 = new Symbol("F1");
var F2 = new Symbol("F2");
var F3 = new Symbol("F3");
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");
var th3 = new Symbol("th3");
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2 };
var _F3 = new Point() { magnitude = F3, angle = th3 };
var m = new Symbol("m");
var obj = new Obj();
obj.acceleration.x = 0;
obj.acceleration.y = 0;
obj.mass = m;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
obj.forces.Add(_F3);
//"Tension in line to kite:".Disp(); "".Disp();
//obj.ForceMagnitude(_F2)
// .Substitute(th1, (180).ToRadians())
// .Substitute(th2, (46.3 * Math.PI / 180))
// .Substitute(th3, (270).ToRadians())
// .Substitute(F3, 0.132 * 9.8)
// .Substitute(Trig.Pi, Math.PI)
// .Disp();
AssertEqual(
obj.ForceMagnitude(_F2)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (46.3 * Math.PI / 180))
.Substitute(th3, (270).ToRadians())
.Substitute(F3, 0.132 * 9.8)
.Substitute(Trig.Pi, Math.PI),
1.7892929261294661);
//Console.ReadLine();
}
#endregion
#region PSE 5E PROBLEM 5.28
{
// A fire helicopter carries a 620-kg bucket of water at the
// end of a cable 20.0 m long. As the aircraft flies back
// from a fire at a constant speed of 40.0 m/s, the cable
// makes an angle of 40.0° with respect to the vertical.
// (a) Determine the force of air resistance on the bucket.
// (b) After filling the bucket with sea water, the pilot re-
// turns to the fire at the same speed with the bucket now
// making an angle of 7.00° with the vertical. What is the
// mass of the water in the bucket?
var F1 = new Symbol("F1"); // force of air resistance
var F2 = new Symbol("F2"); // force of cable
var F3 = new Symbol("F3"); // force of gravity
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");
var th3 = new Symbol("th3");
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2 };
var _F3 = new Point() { magnitude = F3, angle = th3 };
var m = new Symbol("m");
var obj = new Obj();
obj.acceleration.x = 0;
obj.acceleration.y = 0;
obj.mass = m;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
obj.forces.Add(_F3);
//"Force of air resistance on the bucket:".Disp(); "".Disp();
var FAir =
obj.ForceMagnitude(_F1)
.Substitute(F3, 620 * 9.8)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (90 - 40).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI);
AssertEqual(
obj.ForceMagnitude(_F1)
.Substitute(F3, 620 * 9.8)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (90 - 40).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI),
5098.3693590331513);
//"".Disp();
_F1.magnitude = FAir;
_F3.magnitude = null;
var FBucketWithWater =
obj.ForceMagnitude(_F3)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (90 - 7).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI);
//"What is the mass of the water in the bucket?".Disp(); "".Disp();
//(FBucketWithWater / 9.8 - 620).Disp();
AssertEqual(
(FBucketWithWater / 9.8 - 620),
3617.0292120139611);
}
#endregion
#region PSE 5E PROBLEM 5.30
{
// A simple accelerometer is constructed by suspending a
// mass m from a string of length L that is tied to the top
// of a cart. As the cart is accelerated the string-mass
// system makes a constant angle th with the vertical.
// (a) Assuming that the string mass is negligible compared
// with m, derive an expression for the cart’s acceleration
// in terms of and show that it is independent of
// the mass mand the length L.
// (b) Determine the acceleration of the cart when th = 23.0°.
var F1 = new Symbol("F1"); // force of string
var F2 = new Symbol("F2"); // force of gravity
var th1 = new Symbol("th1");
var th2 = new Symbol("th2"); ;
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2, magnitude = F2 };
var m = new Symbol("m");
var g = new Symbol("g");
var obj = new Obj() { mass = m };
obj.acceleration.y = 0;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
_F1.magnitude = obj.ForceMagnitude(_F1);
//("Derive an expression for the cart’s acceleration in terms " +
//"of and show that it is independent of the mass mand the length L:").Disp();
//"".Disp();
//obj.AccelerationX()
// .Substitute(F2, m * g)
// .Substitute(Trig.Cos(th2), 0)
// .Substitute(Trig.Sin(th2), -1)
// .Disp();
//"".Disp();
//"Determine the acceleration of the cart when th = 23.0°".Disp(); "".Disp();
//obj.AccelerationX()
// .Substitute(F2, m * g)
// .Substitute(Trig.Cos(th2), 0)
// .Substitute(Trig.Sin(th2), -1)
// .Substitute(th1, (90 - 23).ToRadians())
// .Substitute(Trig.Pi, Math.PI)
// .Substitute(g, 9.8)
// .Disp();
AssertEqual(
obj.AccelerationX()
.Substitute(F2, m * g)
.Substitute(Trig.Cos(th2), 0)
.Substitute(Trig.Sin(th2), -1)
.Substitute(th1, (90 - 23).ToRadians())
.Substitute(Trig.Pi, Math.PI)
.Substitute(g, 9.8),
4.1598531988541261);
}
#endregion
#region PSE 5E PROBLEM 5.31
{
// Two people pull as hard as they can on ropes attached
// to a boat that has a mass of 200 kg. If they pull in the
// same direction, the boat has an acceleration of
// 1.52 m/s^2 to the right. If they pull in opposite directions,
// the boat has an acceleration of 0.518 m/s^2
// to the left. What is the force exerted by each person on the
// boat? (Disregard any other forces on the boat.)
// Trig.Cos(new DoubleFloat(Math.PI)).Disp();
var m = new Symbol("m");
var aAx = new Symbol("aAx");
var aBx = new Symbol("aBx");
var objA = new Obj() { mass = m };
objA.acceleration.x = aAx;
var _F1A = new Point() { angle = 0 };
var _F2A = new Point() { angle = 0 };
objA.forces.Add(_F1A);
objA.forces.Add(_F2A);
var objB = new Obj() { mass = m };
objB.acceleration.x = aBx;
var _F1B = new Point() { angle = 0 };
var _F2B = new Point() { angle = new DoubleFloat(Math.PI) };
objB.forces.Add(_F1B);
objB.forces.Add(_F2B);
//"force 1 magnitude (symbolic):".Disp(); "".Disp();
//Calc.ForceMagnitude(objA, objB, _F1A, _F1B)
// .Substitute(Trig.Cos(0), 1)
// .Disp();
//"force 1 magnitude (numeric):".Disp(); "".Disp();
//Calc.ForceMagnitude(objA, objB, _F1A, _F1B)
// .Substitute(Trig.Cos(0), 1)
// .Substitute(m, 200)
// .Substitute(aAx, 1.52)
// .Substitute(aBx, -0.518)
// .Disp();
AssertEqual(
Calc.ForceMagnitude(objA, objB, _F1A, _F1B)
.Substitute(Trig.Cos(0), 1)
.Substitute(m, 200)
.Substitute(aAx, 1.52)
.Substitute(aBx, -0.518),
100.19999999999999);
//"".Disp();
//"force 2 magnitude (symbolic):".Disp(); "".Disp();
//Calc.ForceMagnitude(objA, objB, _F2A, _F2B)
// .Substitute(Trig.Cos(0), 1)
// .Disp();
//"force 2 magnitude (numeric):".Disp(); "".Disp();
//Calc.ForceMagnitude(objA, objB, _F2A, _F2B)
// .Substitute(Trig.Cos(0), 1)
// .Substitute(m, 200)
// .Substitute(aAx, 1.52)
// .Substitute(aBx, -0.518)
// .Disp();
//"".Disp();
AssertEqual(
Calc.ForceMagnitude(objA, objB, _F2A, _F2B)
.Substitute(Trig.Cos(0), 1)
.Substitute(m, 200)
.Substitute(aAx, 1.52)
.Substitute(aBx, -0.518),
203.8);
}
#endregion
Console.WriteLine("Testing complete");
Console.ReadLine();
}
static void Main(string[] args)
{
// Three forces, given by _F1 = (-2.00i + 2.00j) N,
// _F2 = (5.00i - 3.00j) N, and _F3 = (-45.0i) N,
// act on an object to give
// it an acceleration of magnitude 3.75 m/s^2.
// (a) What is the direction of the acceleration? (b) What
// is the mass of the object? (c) If the object is initially at
// rest, what is its speed after 10.0 s?
// (d) What are the velocity components of the object after 10.0 s?
var F1x = new Symbol("F1x");
var F1y = new Symbol("F1y");
var F2x = new Symbol("F2x");
var F2y = new Symbol("F2y");
var F3x = new Symbol("F3x");
var F3y = new Symbol("F3y");
var a = new Symbol("a");
var t = new Symbol("t");
var obj = new Obj()
{
forces = new List<Point>()
{
new Point(F1x, F1y),
new Point(F2x, F2y),
new Point(F3x, F3y)
}
};
"What is the direction of the acceleration?".Disp(); "".Disp();
"symbolic:".Disp(); "".Disp();
obj.TotalForce().ToAngle().Disp(); "".Disp();
"numeric:".Disp(); "".Disp();
obj.TotalForce().ToAngle()
.ToDegrees()
.Substitute(F1x, -2)
.Substitute(F1y, 2)
.Substitute(F2x, 5)
.Substitute(F2y, -3)
.Substitute(F3x, -45)
.Substitute(F3y, 0)
.Substitute(Trig.Pi, Math.PI)
.Disp();
"".Disp();
"What is the mass of the object?".Disp(); "".Disp();
obj.totalForce = obj.TotalForce();
obj.acceleration = Point.FromAngle(obj.totalForce.ToAngle(), a);
obj.mass = obj.Mass();
"symbolic:".Disp(); "".Disp();
obj.mass.Disp(); "".Disp();
"numeric:".Disp(); "".Disp();
obj.mass
.Substitute(a, 3.75)
.Substitute(F1x, -2)
.Substitute(F1y, 2)
.Substitute(F2x, 5)
.Substitute(F2y, -3)
.Substitute(F3x, -45)
.Substitute(F3y, 0)
.Disp();
"".Disp();
"If the object is initially at rest, what is its speed after 10.0 s?".Disp();
"".Disp();
obj.velocity = new Point(0, 0);
obj.time = 0;
obj.VelocityAtTime(t).Norm().Disp();
"".Disp();
obj.VelocityAtTime(t).Norm()
.Substitute(a, 3.75)
.Substitute(F1x, -2)
.Substitute(F1y, 2)
.Substitute(F2x, 5)
.Substitute(F2y, -3)
.Substitute(F3x, -45)
.Substitute(F3y, 0)
.Substitute(t, 10)
.Disp();
"".Disp();
"What are the velocity components of the object after 10.0 s?".Disp();
"".Disp();
"velocity.x: ".Disp();
obj.VelocityAtTime(t).x
.Substitute(a, 3.75)
.Substitute(F1x, -2)
.Substitute(F1y, 2)
.Substitute(F2x, 5)
.Substitute(F2y, -3)
.Substitute(F3x, -45)
.Substitute(F3y, 0)
.Substitute(t, 10)
.Disp(); "".Disp();
"velocity.y: ".Disp();
obj.VelocityAtTime(t).y
.Substitute(a, 3.75)
.Substitute(F1x, -2)
.Substitute(F1y, 2)
.Substitute(F2x, 5)
.Substitute(F2y, -3)
.Substitute(F3x, -45)
.Substitute(F3y, 0)
.Substitute(t, 10)
.Disp(); "".Disp();
Console.ReadLine();
}
static void Main(string[] args)
{
// A long-jumper leaves the ground at an angle of 20.0° above
// the horizontal and at a speed of 11.0 m/s.
// (a) How far does he jump in the horizontal direction?
// (Assume his motion is equivalent to that of a particle.)
// (b) What is the maximum height reached?
// For the purposes of solving the problem, let's take
// point A as the initial position
// point B as the peak position
// point C as the final position.
// First we'll get the answer in a symbolic form.
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
// An Obj representing the object at A:
var objA =
new Obj()
{
position = new Point(0, 0),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = 0
};
var objB =
new Obj()
{
velocity = new Point(objA.velocity.x, 0),
acceleration = _g
};
var timeB = Calc.Time(objA, objB);
var timeC = timeB * 2;
objB = objA.AtTime(timeB);
var objC = objA.AtTime(timeC);
Console.WriteLine("How far does he jump in the horizontal direction?");
Console.WriteLine(objC.position.x);
Console.WriteLine();
Console.WriteLine("What is the maximum height reached?");
Console.WriteLine(objB.position.y);
Console.WriteLine();
Console.WriteLine("Now for the numerical solutions:");
Console.WriteLine();
Console.WriteLine("Distance jumped: ");
Console.WriteLine(
objC.position.x
.Substitute(thA, Trig.ToRadians(20))
.Substitute(g, 9.8)
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 11)
);
Console.WriteLine();
Console.WriteLine("Maximum height reached: ");
Console.WriteLine(
objB.position.y
.Substitute(g, 9.8)
.Substitute(thA, Trig.ToRadians(20))
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 11)
);
Console.ReadLine();
}
static void Main(string[] args)
{
// A bag of cement of weight 325 N hangs from three
// wires as shown in Figure P5.24. Two of the wires make
// angles th1 = 60.0° and th2 = 25.0° with the horizontal. If
// the system is in equilibrium, find the tensions
// T1, T2, and T3 in the wires.
// From problem 5.25:
// If the system is in equilibrium, show that
// the tension in the left-hand wire is
//
// T1 = Fg cos(th2) / sin(th1 + th2)
var F3 = new Symbol("F3");
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");
var th3 = new Symbol("th3");
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2 };
var _F3 = new Point() { magnitude = F3, angle = th3 };
var m = new Symbol("m");
var obj = new Obj();
obj.acceleration.x = 0;
obj.acceleration.y = 0;
obj.mass = m;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
obj.forces.Add(_F3);
"F1 magnitude, symbolic:".Disp(); "".Disp();
obj.ForceMagnitude(_F1).Disp(); "".Disp();
"F1 magnitude, numeric:".Disp(); "".Disp();
obj.ForceMagnitude(_F1)
.Substitute(F3, 325)
.Substitute(th1, (180 - 60).ToRadians())
.Substitute(th2, (25).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI)
.Disp();
"".Disp();
"F3 magnitude, numeric:".Disp(); "".Disp();
obj.ForceMagnitude(_F2)
.Substitute(F3, 325)
.Substitute(th1, (180 - 60).ToRadians())
.Substitute(th2, (25).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI)
.Disp();
Console.ReadLine();
}
static void Main(string[] args)
{
// A 15.0-lb block rests on the floor. (a) What force does
// the floor exert on the block? (b) If a rope is tied to the
// block and run vertically over a pulley, and the other end
// is attached to a free-hanging 10.0-lb weight, what is the
// force exerted by the floor on the 15.0-lb block? (c) If we
// replace the 10.0-lb weight in part (b) with a 20.0-lb
// weight, what is the force exerted by the floor on the
// 15.0-lb block?
var thn = new Symbol("thn"); // angle of normal force
var thg = new Symbol("thg"); // angle of gravity force
var th3 = new Symbol("th3");
var Fn = new Symbol("Fn"); // normal force
var Fg = new Symbol("Fg"); // gravity force
var F3 = new Symbol("F3"); // rope force
var m = new Symbol("m");
var ay = new Symbol("ay");
var g = new Symbol("g");
{
var obj = new Obj() { mass = m };
obj.acceleration.y = ay;
obj.forces.Add(new Point() { angle = thn });
obj.forces.Add(new Point() { angle = thg, magnitude = Fg });
"Without rope, normal force magnitude:".Disp(); "".Disp();
obj.ForceMagnitude(obj.forces[0])
.Substitute(thn, (90).ToRadians())
.Substitute(thg, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI)
.Substitute(Fg, 15)
.Substitute(ay, 0)
.Disp();
"".Disp();
}
{
var obj = new Obj() { mass = m };
obj.acceleration.y = ay;
obj.forces.Add(new Point() { angle = thn });
obj.forces.Add(new Point() { angle = thg, magnitude = Fg });
obj.forces.Add(new Point() { angle = th3, magnitude = F3 });
"Rope & 10 lb weight, normal force magnitude:".Disp(); "".Disp();
obj.ForceMagnitude(obj.forces[0])
.Substitute(thn, (90).ToRadians())
.Substitute(thg, (270).ToRadians())
.Substitute(th3, (90).ToRadians())
.Substitute(Trig.Pi, Math.PI)
.Substitute(Fg, 15)
.Substitute(F3, 10)
.Substitute(ay, 0)
.Disp();
}
Console.ReadLine();
}
static void Main(string[] args)
{
// Two forces _F1 and _F2 act on a 5.00-kg mass.
// If F1 = 20.0 N and F2 = 15.0 N, find the accelerations in
// (a) and (b) of Figure P5.15.
var F1 = new Symbol("F1");
var F2 = new Symbol("F2");
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");
var m = new Symbol("m");
var obj = new Obj() { mass = m };
obj.forces.Add(Point.FromAngle(th1, F1));
obj.forces.Add(Point.FromAngle(th2, F2));
"symbolic:".Disp(); "".Disp();
"acceleration.x:".Disp(); "".Disp();
obj.Acceleration().x.Disp(); "".Disp();
"acceleration.y:".Disp(); "".Disp();
obj.Acceleration().y.Disp(); "".Disp();
"numeric (th2 = 90 deg):".Disp(); "".Disp();
Console.Write("acceleration is ");
Console.Write(
obj.Acceleration().Norm()
.Substitute(th1, 0.0)
.Substitute(F1, 20)
.Substitute(th2, (90).ToRadians())
.Substitute(F2, 15)
.Substitute(m, 5)
.Substitute(Trig.Pi, Math.PI));
Console.Write(" at ");
Console.Write(
obj.Acceleration().ToAngle().ToDegrees()
.Substitute(th1, 0.0)
.Substitute(F1, 20)
.Substitute(th2, (90).ToRadians())
.Substitute(F2, 15)
.Substitute(m, 5)
.Substitute(Trig.Pi, Math.PI));
Console.Write(" degrees"); "".Disp(); "".Disp();
"numeric (th2 = 60 deg):".Disp(); "".Disp();
Console.Write("acceleration is ");
Console.Write(
obj.Acceleration().Norm()
.Substitute(th1, 0.0)
.Substitute(F1, 20)
.Substitute(th2, (60).ToRadians())
.Substitute(F2, 15)
.Substitute(m, 5)
.Substitute(Trig.Pi, Math.PI));
Console.Write(" at ");
Console.Write(
obj.Acceleration().ToAngle().ToDegrees()
.Substitute(th1, 0.0)
.Substitute(F1, 20)
.Substitute(th2, (60).ToRadians())
.Substitute(F2, 15)
.Substitute(m, 5)
.Substitute(Trig.Pi, Math.PI));
Console.Write(" degrees"); Console.WriteLine();
Console.ReadLine();
}
static void Main(string[] args)
{
Action<Equation> AssertIsTrue = (eq) =>
{
if (!eq) Console.WriteLine(eq.ToString());
};
Func<MathObject, MathObject> sin = obj => Trig.Sin(obj);
Func<MathObject, MathObject> cos = obj => Trig.Cos(obj);
Func<MathObject, MathObject> tan = obj => new Tan(obj).Simplify();
Func<MathObject, MathObject> asin = obj => new Asin(obj).Simplify();
Func<MathObject, MathObject> atan = obj => new Atan(obj).Simplify();
{
var a = new Symbol("a");
var b = new Symbol("b");
var c = new Symbol("c");
var d = new Symbol("d");
var x = new Symbol("x");
var y = new Symbol("y");
var z = new Symbol("z");
Func<int, Integer> Int = (n) => new Integer(n);
{
DoubleFloat.tolerance = 0.000000001;
Assert(new DoubleFloat(1.2).Equals(new DoubleFloat(1.2)), "new DoubleFloat(1.2).Equals(new DoubleFloat(1.2))");
Assert(new DoubleFloat(1.20000001).Equals(new DoubleFloat(1.20000002)) == false, "new DoubleFloat(1.20000001).Equals(new DoubleFloat(1.20000002)) == false");
Assert(new DoubleFloat(1.2000000000001).Equals(new DoubleFloat(1.200000000002)), "new DoubleFloat(1.2000000000001).Equals(new DoubleFloat(1.200000000002))");
Assert(new DoubleFloat(1.2).Equals(new DoubleFloat(1.23)) == false, "new DoubleFloat(1.2).Equals(new DoubleFloat(1.23)) == false");
DoubleFloat.tolerance = null;
}
#region Simplify
AssertIsTrue(x + x == 2 * x);
AssertIsTrue(x + x == 2 * x);
AssertIsTrue(x + x + x == 3 * x);
AssertIsTrue(5 + x + 2 == 7 + x);
AssertIsTrue(3 + x + 5 + x == 8 + 2 * x);
AssertIsTrue(4 * x + 3 * x == 7 * x);
AssertIsTrue(x + y + z + x + y + z == 2 * x + 2 * y + 2 * z);
AssertIsTrue(10 - x == 10 + x * -1);
AssertIsTrue(x * y / 3 == Int(1) / 3 * x * y);
AssertIsTrue(x / y == x * (y ^ -1));
AssertIsTrue(x / 3 == x * (Int(1) / 3));
AssertIsTrue(6 * x * y / 3 == 2 * x * y);
AssertIsTrue((((x ^ Int(1) / 2) ^ Int(1) / 2) ^ 8) == (x ^ 2));
AssertIsTrue(((((x * y) ^ (Int(1) / 2)) * (z ^ 2)) ^ 2) == (x * y * (z ^ 4)));
AssertIsTrue(x / x == 1);
AssertIsTrue(x / y * y / x == 1);
AssertIsTrue((x ^ 2) * (x ^ 3) == (x ^ 5));
AssertIsTrue(x + y + x + z + 5 + z == 5 + 2 * x + y + 2 * z);
AssertIsTrue(((Int(1) / 2) * x + (Int(3) / 4) * x) == Int(5) / 4 * x);
AssertIsTrue(1.2 * x + 3 * x == 4.2 * x);
AssertIsTrue(3 * x + 1.2 * x == 4.2 * x);
AssertIsTrue(1.2 * x * 3 * y == 3.5999999999999996 * x * y);
AssertIsTrue(3 * x * 1.2 * y == 3.5999999999999996 * x * y);
AssertIsTrue(3.4 * x * 1.2 * y == 4.08 * x * y);
AssertIsTrue((a == b) == (a == b));
#endregion
#region Power.Simplify
AssertIsTrue((0 ^ x) == 0);
AssertIsTrue((1 ^ x) == 1);
AssertIsTrue((x ^ 0) == 1);
AssertIsTrue((x ^ 1) == x);
#endregion
// Product.Simplify
AssertIsTrue(x * 0 == 0);
// Difference
AssertIsTrue(-x == -1 * x);
AssertIsTrue(x - y == x + -1 * y);
#region Substitute
AssertIsTrue(Int(10).Substitute(Int(10), 20) == 20);
AssertIsTrue(Int(10).Substitute(Int(15), 20) == 10);
AssertIsTrue(new DoubleFloat(1.0).Substitute(new DoubleFloat(1.0), 2.0) == 2.0);
AssertIsTrue(new DoubleFloat(1.0).Substitute(new DoubleFloat(1.5), 2.0) == 1.0);
AssertIsTrue((Int(1) / 2).Substitute(Int(1) / 2, Int(3) / 4) == Int(3) / 4);
AssertIsTrue((Int(1) / 2).Substitute(Int(1) / 3, Int(3) / 4) == Int(1) / 2);
AssertIsTrue(x.Substitute(x, y) == y);
AssertIsTrue(x.Substitute(y, y) == x);
AssertIsTrue((x ^ y).Substitute(x, 10) == (10 ^ y));
AssertIsTrue((x ^ y).Substitute(y, 10) == (x ^ 10));
AssertIsTrue((x ^ y).Substitute(x ^ y, 10) == 10);
AssertIsTrue((x * y * z).Substitute(x, y) == ((y ^ 2) * z));
AssertIsTrue((x * y * z).Substitute(x * y * z, x) == x);
AssertIsTrue((x + y + z).Substitute(x, y) == ((y * 2) + z));
AssertIsTrue((x + y + z).Substitute(x + y + z, x) == x);
AssertIsTrue(
((((x * y) ^ (Int(1) / 2)) * (z ^ 2)) ^ 2)
.Substitute(x, 10)
.Substitute(y, 20)
.Substitute(z, 3)
== 16200
);
#region Equation.Substitute
AssertIsTrue((x == y).Substitute(y, z) == (x == z));
AssertIsTrue((x != y).Substitute(y, z) == (x != z));
(x == 0).Substitute(x, 0).AssertEqTo(true);
(x == 0).Substitute(x, 1).AssertEqTo(false);
(x != 0).Substitute(x, 0).AssertEqTo(false);
(x != 0).Substitute(x, 1).AssertEqTo(true);
#endregion
#endregion
AssertIsTrue(sin(new DoubleFloat(3.14159 / 2)) == 0.99999999999911982);
AssertIsTrue(sin(x + y) + sin(x + y) == 2 * sin(x + y));
AssertIsTrue(sin(x + x) == sin(2 * x));
AssertIsTrue(sin(x + x).Substitute(x, 1) == sin(Int(2)));
AssertIsTrue(sin(x + x).Substitute(x, 1.0) == 0.90929742682568171);
AssertIsTrue(sin(2 * x).Substitute(x, y) == sin(2 * y));
// Product.RecursiveSimplify
AssertIsTrue(1 * x == x);
AssertIsTrue(x * 1 == x);
AssertIsTrue(x != y);
AssertIsTrue(x != 10);
// ==(double a, MathObject b)
AssertIsTrue(1.0 == new DoubleFloat(3.0) - 2.0);
AssertIsTrue((a == b) != (a != b));
#region Equation.ToString
Assert((x == y).ToString() == "x == y", "x == y");
Assert((x != y).ToString() == "x != y", "x != y");
#endregion
#region Function.ToString
Assert(new And().ToString() == "And()", "And()");
#endregion
#region Equation.Simplify
(new Integer(0) == new Integer(0)).Simplify().AssertEqTo(true);
(new Integer(0) == new Integer(1)).Simplify().AssertEqTo(false);
(new Integer(0) != new Integer(1)).Simplify().AssertEqTo(true);
(new Integer(0) != new Integer(0)).Simplify().AssertEqTo(false);
#endregion
#region And
new And().Simplify().AssertEqTo(true);
new And(10).Simplify().AssertEqTo(10);
new And(true).Simplify().AssertEqTo(true);
new And(false).Simplify().AssertEqTo(false);
new And(10, 20, 30).Simplify().AssertEqTo(new And(10, 20, 30));
new And(10, false, 20).Simplify().AssertEqTo(false);
new And(10, true, 20).Simplify().AssertEqTo(new And(10, 20));
new And(10, new And(20, 30), 40)
.Simplify()
.AssertEqTo(new And(10, 20, 30, 40));
#endregion
#region Or
new Or(10).Simplify().AssertEqTo(10);
new Or(true).Simplify().AssertEqTo(true);
new Or(false).Simplify().AssertEqTo(false);
new Or(10, 20, false).Simplify().AssertEqTo(new Or(10, 20));
new Or(false, false).Simplify().AssertEqTo(false);
new Or(10, true, 20, false).Simplify().AssertEqTo(true);
new Or(10, false, 20).Simplify().AssertEqTo(new Or(10, 20));
new Or(10, new Or(20, 30), 40)
.Simplify()
.AssertEqTo(new Or(10, 20, 30, 40));
#endregion
#region Function.Map
new And(1, 2, 3, 4, 5, 6).Map(elt => elt * 2)
.AssertEqTo(new And(2, 4, 6, 8, 10, 12));
new And(1, 2, 3, 4, 5, 6).Map(elt => (elt is Integer) && (elt as Integer).val % 2 == 0 ? elt : false)
.AssertEqTo(false);
new Or(1, 2, 3).Map(elt => elt * 2)
.AssertEqTo(new Or(2, 4, 6));
new Or(1, 2, 3, 4, 5, 6).Map(elt => (elt is Integer) && (elt as Integer).val % 2 == 0 ? elt : false)
.AssertEqTo(new Or(2, 4, 6));
#endregion Function.Map
#region Sum
Assert((x + y).Equals(x * y) == false, "(x + y).Equals(x * y)");
#endregion
#region Has
Assert(a.Has(elt => elt == a), "a.Has(elt => elt == a)");
Assert(a.Has(elt => elt == b) == false, "a.Has(elt => elt == b) == false");
Assert((a == b).Has(elt => elt == a), "Has - 3");
Assert((a == b).Has(elt => elt == c) == false, "Has - 4");
Assert(((a + b) ^ c).Has(elt => elt == a + b), "Has - 5");
Assert(((a + b) ^ c).Has(elt => (elt is Power) && (elt as Power).exp == c), "Has - 6");
Assert((x * (a + b + c)).Has(elt => (elt is Sum) && (elt as Sum).Has(b)), "Has - 7");
Assert((x * (a + b + c)).Has(elt => (elt is Sum) && (elt as Sum).elts.Any(obj => obj == b)), "Has - 8");
Assert((x * (a + b + c)).Has(elt => (elt is Product) && (elt as Product).elts.Any(obj => obj == b)) == false, "Has - 9");
#endregion
#region FreeOf
Assert((a + b).FreeOf(b) == false, "(a + b).FreeOf(b)");
Assert((a + b).FreeOf(c) == true, "(a + b).FreeOf(c)");
Assert(((a + b) * c).FreeOf(a + b) == false, "((a + b) * c).FreeOf(a + b)");
Assert((sin(x) + 2 * x).FreeOf(sin(x)) == false, "(sin(x) + 2 * x).FreeOf(sin(x))");
Assert(((a + b + c) * d).FreeOf(a + b) == true, "((a + b + c) * d).FreeOf(a + b)");
Assert(((y + 2 * x - y) / x).FreeOf(x) == true, "((y + 2 * x - y) / x).FreeOf(x)");
Assert(((x * y) ^ 2).FreeOf(x * y) == true, "((x * y) ^ 2).FreeOf(x * y)");
#endregion
#region Denominator
{
((new Integer(2) / 3) * ((x * (x + 1)) / (x + 2)) * (y ^ z))
.Denominator()
.AssertEqTo(3 * (x + 2));
}
#endregion
#region LogicalExpand
new And(new Or(a, b), c)
.LogicalExpand()
.AssertEqTo(
new Or(
new And(a, c),
new And(b, c)));
new And(a, new Or(b, c))
.LogicalExpand()
.AssertEqTo(new Or(new And(a, b), new And(a, c)));
new And(a, new Or(b, c), d)
.LogicalExpand()
.AssertEqTo(
new Or(
new And(a, b, d),
new And(a, c, d)));
new And(new Or(a == b, b == c), x == y)
.LogicalExpand()
.AssertEqTo(
new Or(
new And(a == b, x == y),
new And(b == c, x == y)));
new And(
new Or(a == b, b == c),
new Or(c == d, d == a),
x == y)
.LogicalExpand()
.AssertEqTo(
new Or(
new And(a == b, c == d, x == y),
new And(a == b, d == a, x == y),
new And(b == c, c == d, x == y),
new And(b == c, d == a, x == y)));
#endregion
#region SimplifyEquation
(2 * x == 0)
.SimplifyEquation()
.AssertEqTo(x == 0);
(2 * x != 0)
.SimplifyEquation()
.AssertEqTo(x != 0);
((x ^ 2) == 0)
.SimplifyEquation()
.AssertEqTo(x == 0);
#endregion
#region SimplifyLogical
new And(a, b, c, a)
.SimplifyLogical()
.AssertEqTo(new And(a, b, c));
#endregion SimplifyLogical
#region DegreeGpe
{
var w = new Symbol("w");
Assert(
((3 * w * x ^ 2) * (y ^ 3) * (z ^ 4)).DegreeGpe(new List<MathObject>() { x, z }) == 6,
"((3 * w * x ^ 2) * (y ^ 3) * (z ^ 4)).DegreeGpe(new List<MathObject>() { x, z })");
Assert(
((a * x ^ 2) + b * x + c).DegreeGpe(new List<MathObject>() { x }) == 2,
"((a * x ^ 2) + b * x + c).DegreeGpe(new List<MathObject>() { x })");
Assert(
(a * (sin(x) ^ 2) + b * sin(x) + c).DegreeGpe(new List<MathObject>() { sin(x) }) == 2,
"(a * (sin(x) ^ 2) + b * sin(x) + c).DegreeGpe(new List<MathObject>() { sin(x) })");
Assert(
(2 * (x ^ 2) * y * (z ^ 3) + w * x * (z ^ 6)).DegreeGpe(new List<MathObject>() { x, z }) == 7,
"(2 * (x ^ 2) * y * (z ^ 3) + w * x * (z ^ 6)).DegreeGpe(new List<MathObject>() { x, z })");
}
#endregion
#region CoefficientGpe
AssertIsTrue((a * (x ^ 2) + b * x + c).CoefficientGpe(x, 2) == a);
AssertIsTrue((3 * x * (y ^ 2) + 5 * (x ^ 2) * y + 7 * x + 9).CoefficientGpe(x, 1) == 3 * (y ^ 2) + 7);
AssertIsTrue((3 * x * (y ^ 2) + 5 * (x ^ 2) * y + 7 * x + 9).CoefficientGpe(x, 3) == 0);
Assert(
(3 * sin(x) * (x ^ 2) + 2 * x + 4).CoefficientGpe(x, 2) == null,
"(3 * sin(x) * (x ^ 2) + 2 * x + 4).CoefficientGpe(x, 2) == null");
#endregion
#region AlgebraicExpand
AssertIsTrue(
((x + 2) * (x + 3) * (x + 4)).AlgebraicExpand()
==
24 + 26 * x + 9 * (x ^ 2) + (x ^ 3));
AssertIsTrue(
((x + y + z) ^ 3).AlgebraicExpand()
==
(x ^ 3) + (y ^ 3) + (z ^ 3) +
3 * (x ^ 2) * y +
3 * (y ^ 2) * x +
3 * (x ^ 2) * z +
3 * (y ^ 2) * z +
3 * (z ^ 2) * x +
3 * (z ^ 2) * y +
6 * x * y * z);
AssertIsTrue(
(((x + 1) ^ 2) + ((y + 1) ^ 2)).AlgebraicExpand()
==
2 + 2 * x + (x ^ 2) + 2 * y + (y ^ 2));
AssertIsTrue(
((((x + 2) ^ 2) + 3) ^ 2).AlgebraicExpand()
==
49 + 56 * x + 30 * (x ^ 2) + 8 * (x ^ 3) + (x ^ 4));
AssertIsTrue(
sin(x * (y + z)).AlgebraicExpand()
==
sin(x * y + x * z));
AssertIsTrue(
(a * (b + c) == x * (y + z)).AlgebraicExpand()
==
(a * b + a * c == x * y + x * z));
#endregion
#region IsolateVariable
(x + y + z == 0).IsolateVariable(a).AssertEqTo(x + y + z == 0);
// (x * a + x * b == 0).IsolateVariable(x).Disp();
(x * (a + b) - x * a - x * b + x == c)
.IsolateVariable(x)
.AssertEqTo(x == c);
new And(x == y, a == b)
.IsolateVariable(b)
.AssertEqTo(new And(x == y, b == a));
new Or(new And(y == x, z == x), new And(b == x, c == x))
.IsolateVariable(x)
.AssertEqTo(new Or(new And(x == y, x == z), new And(x == b, x == c)));
Assert((0 == x - y).IsolateVariableEq(x).Equals(x == y), "(0 == x - y).IsolateVariable(x).Equals(x == y)");
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
(a * (x ^ 2) + b * x + c == 0)
.IsolateVariable(x)
.AssertEqTo(
new Or(
new And(
x == (-b + sqrt((b ^ 2) + -4 * a * c)) / (2 * a),
a != 0
),
new And(
x == (-b - sqrt((b ^ 2) + -4 * a * c)) / (2 * a),
a != 0
),
new And(x == -c / b, a == 0, b != 0),
new And(a == 0, b == 0, c == 0)
)
);
(a * (x ^ 2) + c == 0)
.IsolateVariable(x)
.AssertEqTo(
new Or(
new And(
x == sqrt(-4 * a * c) / (2 * a),
a != 0
),
new And(
x == -sqrt(-4 * a * c) / (2 * a),
a != 0
),
new And(a == 0, c == 0)
)
);
// a x^2 + b x + c == 0
// a x^2 + c == - b x
// (a x^2 + c) / x == - b
((a * (x ^ 2) + c) / x == -b)
.IsolateVariable(x)
.AssertEqTo(
new Or(
new And(
x == (-b + sqrt((b ^ 2) + -4 * a * c)) / (2 * a),
a != 0
),
new And(
x == (-b - sqrt((b ^ 2) + -4 * a * c)) / (2 * a),
a != 0
),
new And(x == -c / b, a == 0, b != 0),
new And(a == 0, b == 0, c == 0)
)
);
(sqrt(x + y) == z).IsolateVariable(x).AssertEqTo(x == (z ^ 2) - y);
(a * b + a == c)
.IsolateVariable(a)
.AssertEqTo(a == c / (b + 1));
(a * b + a * c == d)
.IsolateVariable(a)
.AssertEqTo(a == d / (b + c));
(1 / sqrt(x) == y)
.IsolateVariable(x)
.AssertEqTo(x == (y ^ -2));
(y == sqrt(x) / x)
.IsolateVariable(x)
.AssertEqTo(x == (y ^ -2));
(-sqrt(x) + z * x == y)
.IsolateVariable(x)
.AssertEqTo(-sqrt(x) + z * x == y);
(sqrt(a + x) - z * x == -y)
.IsolateVariable(x)
.AssertEqTo(sqrt(a + x) - z * x == -y);
(sqrt(2 + x) * sqrt(3 + x) == y)
.IsolateVariable(x)
.AssertEqTo(sqrt(2 + x) * sqrt(3 + x) == y);
#endregion
#region EliminateVariable
new And((x ^ 3) == (y ^ 5), z == x)
.EliminateVariable(x)
.AssertEqTo((z ^ 3) == (y ^ 5));
new And((x ^ 3) == (y ^ 5), z == (x ^ 7))
.EliminateVariable(x)
.AssertEqTo(new And((x ^ 3) == (y ^ 5), z == (x ^ 7)));
#endregion
}
#region EliminateVariable
#region
{
var x = new Symbol("x");
var y = new Symbol("y");
var z = new Symbol("z");
var eqs = new And()
{
args =
{
(x ^ 2) - 4 == 0,
y + x == 0,
x + z == 10
}
}
.Simplify();
var half = new Integer(1) / 2;
Func<MathObject, MathObject> sqrt = obj => obj ^ half;
((x ^ 2) - 4 == 0)
.IsolateVariableEq(x)
.AssertEqTo(new Or(x == half * sqrt(16), x == -half * sqrt(16)));
eqs.EliminateVariable(x)
.AssertEqTo(
new Or(
new And(
half * sqrt(16) + y == 0,
half * sqrt(16) + z == 10
),
new And(
-half * sqrt(16) + y == 0,
-half * sqrt(16) + z == 10
)
)
);
}
#endregion
#region
{
var a = new Symbol("a");
var x = new Symbol("x");
var y = new Symbol("y");
var z = new Symbol("z");
new Or(
new And(x == y, x == z, x == a),
new And(x == -y, x == z, x == a)
)
.EliminateVariable(x)
.AssertEqTo(
new Or(
new And(y == z, y == a),
new And(-y == z, -y == a)
)
)
.EliminateVariable(y)
.AssertEqTo(new Or(z == a, z == a));
}
#endregion
{
var x = new Symbol("x");
var y = new Symbol("y");
var z = new Symbol("z");
new And(y != z, y == x, y == 10)
.EliminateVariable(y)
.AssertEqTo(new And(x != z, x == 10));
}
#region PSE Example 2.6
{
var sAC = new Symbol("sAC");
var sAB = new Symbol("sAB");
var vA = new Symbol("vA");
var vB = new Symbol("vB");
var vC = new Symbol("vC");
var a = new Symbol("a");
var tAC = new Symbol("tAC");
var tAB = new Symbol("tAB");
var eqs = new And(tAB == tAC / 2);
eqs.args.AddRange(Kinematic(sAC, vA, vC, a, tAC));
eqs.args.AddRange(Kinematic(sAB, vA, vB, a, tAB));
var vals = new List<Equation>() { vA == 10, vC == 30, tAC == 10 };
eqs
.EliminateVariables(tAB, sAC, vB, sAB)
.IsolateVariable(a)
.AssertEqTo(a == (vC - vA) / tAC)
.SubstituteEqLs(vals)
.AssertEqTo(a == 2);
eqs
.EliminateVariables(vB, a, tAB, sAC)
.AssertEqTo(sAB == tAC / 4 * (2 * vA + (vC - vA) / 2))
.SubstituteEqLs(vals)
.AssertEqTo(sAB == 75);
}
#endregion
#region PSE Example 2.7
{
// s =
// u = 63
// v = 0
// a =
// t = 2
var s = new Symbol("s");
var u = new Symbol("u");
var v = new Symbol("v");
var a = new Symbol("a");
var t = new Symbol("t");
var eqs = new And();
eqs.args.AddRange(Kinematic(s, u, v, a, t));
var vals = new List<Equation>() { u == 63, v == 0, t == 2.0 };
eqs
.EliminateVariable(s)
.AssertEqTo(v == a * t + u)
.IsolateVariable(a)
.AssertEqTo(a == (v - u) / t)
.SubstituteEqLs(vals)
.AssertEqTo(a == -31.5);
eqs
.EliminateVariable(a)
.SubstituteEqLs(vals)
.AssertEqTo(s == 63.0);
}
#endregion
#region PSE Example 2.8
{
// car
//
// s1 =
// u1 = 45
// v1 = 45
// a1 = 0
// t1 =
// officer
//
// s2 =
// u2 = 0
// v2 =
// a2 = 3
// t2
var s1 = new Symbol("s1");
var u1 = new Symbol("u1");
var v1 = new Symbol("v1");
var a1 = new Symbol("a1");
var t1 = new Symbol("t1");
var s2 = new Symbol("s2");
var u2 = new Symbol("u2");
var v2 = new Symbol("v2");
var a2 = new Symbol("a2");
var t2 = new Symbol("t2");
var eqs = new And(
u1 == v1,
s1 == s2,
t2 == t1 - 1);
eqs.args.AddRange(Kinematic(s1, u1, v1, a1, t1));
eqs.args.AddRange(Kinematic(s2, u2, v2, a2, t2));
var vals = new List<Equation>()
{
v1 == 45.0,
u2 == 0,
a2 == 3
};
eqs
.EliminateVariables(s2, t1, a1, s1, v2, u1)
.IsolateVariable(t2)
.SubstituteEqLs(vals)
.AssertEqTo(new Or(t2 == -0.96871942267131317, t2 == 30.968719422671313));
}
#endregion
#region PSE Example 2.12
{
var yA = new Symbol("yA");
var yB = new Symbol("yB");
var yC = new Symbol("yC");
var yD = new Symbol("yD");
var tA = new Symbol("tA");
var tB = new Symbol("tB");
var tC = new Symbol("tC");
var tD = new Symbol("tD");
var vA = new Symbol("vA");
var vB = new Symbol("vB");
var vC = new Symbol("vC");
var vD = new Symbol("vD");
var a = new Symbol("a");
var eqs = new And();
eqs.args.AddRange(Kinematic(yA, yB, vA, vB, a, tA, tB));
eqs.args.AddRange(Kinematic(yB, yC, vB, vC, a, tB, tC));
eqs.args.AddRange(Kinematic(yC, yD, vC, vD, a, tC, tD));
var vals = new List<Equation>()
{
yA == 50,
yC == 50,
vA == 20,
vB == 0,
a == -9.8,
tA == 0,
tD == 5
};
// velocity and position at t = 5.00 s
DoubleFloat.tolerance = 0.000000001;
eqs
.EliminateVariables(tB, tC, vC, yB, yD)
.SubstituteEqLs(vals)
.AssertEqTo(new Or(vD == -29.000000000000004, vD == -29.000000000000007));
eqs
.EliminateVariables(tB, tC, vC, yB, vD)
.IsolateVariable(yD)
.SubstituteEqLs(vals)
.AssertEqTo(new Or(yD == 27.499999999, yD == 27.499999999));
DoubleFloat.tolerance = null;
}
#endregion
#region PSE Example 4.3
{
var xA = new Symbol("xA");
var xB = new Symbol("xB");
var xC = new Symbol("xC");
var yA = new Symbol("yA");
var yB = new Symbol("yB");
var yC = new Symbol("yC");
var vxA = new Symbol("vxA");
var vxB = new Symbol("vxB");
var vxC = new Symbol("vxC");
var vyA = new Symbol("vyA");
var vyB = new Symbol("vyB");
var vyC = new Symbol("vyC");
var tAB = new Symbol("tAB");
var tAC = new Symbol("tAC");
var ax = new Symbol("ax");
var ay = new Symbol("ay");
var vA = new Symbol("vA");
var thA = new Symbol("thA");
var eqs = new And(
vxA == vA * cos(thA),
vyA == vA * sin(thA),
tAC == 2 * tAB,
vxB == vxA + ax * tAB,
vyB == vyA + ay * tAB,
xB == xA + vxA * tAB + ax * (tAB ^ 2) / 2,
yB == yA + vyA * tAB + ay * (tAB ^ 2) / 2,
vxC == vxA + ax * tAB,
vyC == vyA + ay * tAB,
xC == xA + vxA * tAC + ax * (tAC ^ 2) / 2,
yC == yA + vyA * tAC + ay * (tAC ^ 2) / 2
);
var zeros = new List<Equation>() { xA == 0, yA == 0, ax == 0, vyB == 0 };
var vals = new List<Equation>() { thA == (20).ToRadians(), vA == 11.0, ay == -9.8, Trig.Pi == Math.PI };
eqs
.EliminateVariables(xB, yC, vxB, vxC, vyC, yB, tAC, vxA, vyA, tAB)
.SubstituteEqLs(zeros)
.AssertEqTo(xC == -2 * cos(thA) * sin(thA) * (vA ^ 2) / ay)
.SubstituteEqLs(vals)
.AssertEqTo(xC == 7.9364592624562507);
eqs
.EliminateVariables(xB, yC, vxB, vxC, vyC, xC, vxA, tAC, vyA, tAB)
.SubstituteEqLs(zeros)
.AssertEqTo(yB == -(sin(thA) ^ 2) * (vA ^ 2) / (2 * ay))
.SubstituteEqLs(vals)
.AssertEqTo(yB == 0.72215873425009314);
}
#endregion
#region PSE 5E Example 4.5
{
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
var xA = new Symbol("xA");
var xB = new Symbol("xB");
var xC = new Symbol("xC");
var yA = new Symbol("yA");
var yB = new Symbol("yB");
var yC = new Symbol("yC");
var vxA = new Symbol("vxA");
var vxB = new Symbol("vxB");
var vxC = new Symbol("vxC");
var vyA = new Symbol("vyA");
var vyB = new Symbol("vyB");
var vyC = new Symbol("vyC");
var tAB = new Symbol("tAB");
var tAC = new Symbol("tAC");
var ax = new Symbol("ax");
var ay = new Symbol("ay");
var vA = new Symbol("vA");
var thA = new Symbol("thA");
var vC = new Symbol("vC");
var eqs = new And(
vxA == vA * cos(thA),
vyA == vA * sin(thA),
// tAC == 2 * tAB,
// vxB == vxA + ax * tAB,
// vyB == vyA + ay * tAB,
// xB == xA + vxA * tAB + ax * (tAB ^ 2) / 2,
// yB == yA + vyA * tAB + ay * (tAB ^ 2) / 2,
vxC == vxA + ax * tAC,
vyC == vyA + ay * tAC,
// xC == xA + vxA * tAC + ax * (tAC ^ 2) / 2,
yC == yA + vyA * tAC + ay * (tAC ^ 2) / 2,
vC == sqrt((vxC ^ 2) + (vyC ^ 2)),
ay != 0
);
var zeros = new List<Equation>() { ax == 0, yC == 0 };
var vals = new List<Equation>() { yA == 45, vA == 20, thA == (30).ToRadians(), ay == -9.8, Trig.Pi == Math.PI };
DoubleFloat.tolerance = 0.00001;
eqs
.EliminateVariables(vC, vxA, vxC, vyC, vyA)
.IsolateVariable(tAC)
.LogicalExpand().SimplifyEquation().SimplifyLogical()
.CheckVariable(ay)
.AssertEqTo(
new Or(
new And(
tAC == -(sin(thA) * vA + sqrt((sin(thA) ^ 2) * (vA ^ 2) + 2 * ay * (yC - yA))) / ay,
ay != 0),
new And(
tAC == -(sin(thA) * vA - sqrt((sin(thA) ^ 2) * (vA ^ 2) + 2 * ay * (yC - yA))) / ay,
ay != 0)))
.SubstituteEqLs(zeros)
.SubstituteEqLs(vals)
.AssertEqTo(new Or(tAC == 4.2180489012229376, tAC == -2.1772325746923267));
eqs
.SubstituteEqLs(zeros)
.EliminateVariables(vxC, vxA, vyA, vyC, tAC)
.SimplifyEquation().SimplifyLogical()
.CheckVariable(ay)
.AssertEqTo(
new Or(
new And(
ay != 0,
vC == sqrt((cos(thA) ^ 2) * (vA ^ 2) + ((sin(thA) * vA - (sin(thA) * vA + sqrt((sin(thA) ^ 2) * (vA ^ 2) + -2 * ay * yA))) ^ 2))),
new And(
ay != 0,
vC == sqrt((cos(thA) ^ 2) * (vA ^ 2) + ((sin(thA) * vA - (sin(thA) * vA - sqrt((sin(thA) ^ 2) * (vA ^ 2) + -2 * ay * yA))) ^ 2)))))
.SubstituteEqLs(vals)
.AssertEqTo(new Or(vC == 35.805027579936315, vC == 35.805027579936322));
DoubleFloat.tolerance = null;
}
#endregion
#region PSE 5E Example 4.6
{
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
var xA = new Symbol("xA");
var xB = new Symbol("xB");
var yA = new Symbol("yA");
var yB = new Symbol("yB");
var vxA = new Symbol("vxA");
var vxB = new Symbol("vxB");
var vyA = new Symbol("vyA");
var vyB = new Symbol("vyB");
var tAB = new Symbol("tAB");
var ax = new Symbol("ax");
var ay = new Symbol("ay");
var eqs = new And(
vxB == vxA + ax * tAB,
vyB == vyA + ay * tAB,
xB == xA + vxA * tAB + ax * (tAB ^ 2) / 2,
yB == yA + vyA * tAB + ay * (tAB ^ 2) / 2,
vxA != 0,
ay != 0
);
var vals = new List<Equation>() { xA == 0, yA == 100, vxA == 40, vyA == 0, yB == 0, ax == 0, ay == -9.8, Trig.Pi == Math.PI };
var zeros = vals.Where(eq => eq.b == 0).ToList();
DoubleFloat.tolerance = 0.00001;
eqs
.EliminateVariables(vxB, vyB, tAB)
.IsolateVariable(xB)
.LogicalExpand().SimplifyEquation()
.CheckVariable(ay)
.CheckVariable(vxA).SimplifyLogical()
.SubstituteEq(ax == 0)
.AssertEqTo(
new Or(
new And(
vxA != 0,
xB == -1 * (ay ^ -1) * (vxA ^ 2) * (-1 * (-1 * (vxA ^ -1) * vyA + ay * (vxA ^ -2) * xA) + sqrt(((-1 * (vxA ^ -1) * vyA + ay * (vxA ^ -2) * xA) ^ 2) + 2 * ay * (vxA ^ -2) * ((vxA ^ -1) * vyA * xA - ay / 2 * (vxA ^ -2) * (xA ^ 2) + -1 * yA + yB))),
ay * (vxA ^ -2) != 0,
ay != 0),
new And(
vxA != 0,
xB == -1 * (ay ^ -1) * (vxA ^ 2) * (-1 * (-1 * (vxA ^ -1) * vyA + ay * (vxA ^ -2) * xA) + -1 * sqrt(((-1 * (vxA ^ -1) * vyA + ay * (vxA ^ -2) * xA) ^ 2) + 2 * ay * (vxA ^ -2) * ((vxA ^ -1) * vyA * xA - ay / 2 * (vxA ^ -2) * (xA ^ 2) + -1 * yA + yB))),
ay * (vxA ^ -2) != 0,
ay != 0)))
.SubstituteEqLs(zeros)
.AssertEqTo(
new Or(
new And(
vxA != 0,
xB == -1 / ay * (vxA ^ 2) * sqrt(-2 * ay * (vxA ^ -2) * yA),
ay / (vxA ^ 2) != 0,
ay != 0),
new And(
vxA != 0,
xB == 1 / ay * (vxA ^ 2) * sqrt(-2 * ay * (vxA ^ -2) * yA),
ay / (vxA ^ 2) != 0,
ay != 0)))
.SubstituteEqLs(vals)
.AssertEqTo(new Or(xB == 180.70158058105022, xB == -180.70158058105022));
eqs
.EliminateVariables(vxB, xB, tAB)
.IsolateVariable(vyB)
.LogicalExpand().SimplifyEquation()
.CheckVariable(ay)
.AssertEqTo(
new Or(
new And(
vyB == -1 * ay * sqrt(2 * (ay ^ -1) * ((ay ^ -1) / 2 * (vyA ^ 2) + -1 * yA + yB)),
// (ay ^ -1) != 0,
vxA != 0,
ay != 0),
new And(
vyB == ay * sqrt(2 * (ay ^ -1) * ((ay ^ -1) / 2 * (vyA ^ 2) + -1 * yA + yB)),
// (ay ^ -1) != 0,
vxA != 0,
ay != 0)))
.SubstituteEqLs(zeros)
.AssertEqTo(
new Or(
new And(
vyB == -ay * sqrt(-2 / ay * yA),
// 1 / ay != 0,
vxA != 0,
ay != 0),
new And(
vyB == ay * sqrt(-2 / ay * yA),
// 1 / ay != 0,
vxA != 0,
ay != 0)))
.SubstituteEqLs(vals)
.AssertEqTo(new Or(vyB == 44.271887242357309, vyB == -44.271887242357309));
DoubleFloat.tolerance = null;
}
#endregion
#region PSE 5E Example 4.7
{
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
var xA = new Symbol("xA");
var yA = new Symbol("yA");
var xB = new Symbol("xB");
var yB = new Symbol("yB");
var vxA = new Symbol("vxA");
var vyA = new Symbol("vyA");
var vxB = new Symbol("vxB");
var vyB = new Symbol("vyB");
var tAB = new Symbol("tAB");
var ax = new Symbol("ax");
var ay = new Symbol("ay");
var th = new Symbol("th");
var d = new Symbol("d");
var eqs = new And(
cos(th) == (xB - xA) / d,
sin(th) == (yA - yB) / d,
vxB == vxA + ax * tAB,
vyB == vyA + ay * tAB,
xB == xA + vxA * tAB + ax * (tAB ^ 2) / 2,
yB == yA + vyA * tAB + ay * (tAB ^ 2) / 2,
yB != 0,
ay != 0
);
var vals = new List<Equation>() { xA == 0, yA == 0, vxA == 25, vyA == 0, ax == 0, ay == -9.8, th == (35).ToRadians(), Trig.Pi == Math.PI };
var zeros = vals.Where(eq => eq.b == 0).ToList();
DoubleFloat.tolerance = 0.00001;
eqs
.SubstituteEqLs(zeros)
.EliminateVariables(vxB, vyB, d, yB, tAB)
.IsolateVariable(xB)
.LogicalExpand()
.CheckVariable(ay)
.SimplifyEquation()
.AssertEqTo(
new Or(
new And(
xB == -(sin(th) / cos(th) + sqrt((cos(th) ^ -2) * (sin(th) ^ 2))) * (vxA ^ 2) / ay,
ay / (vxA ^ 2) != 0,
sin(th) / cos(th) * xB != 0,
ay != 0),
new And(
xB == -(sin(th) / cos(th) - sqrt((cos(th) ^ -2) * (sin(th) ^ 2))) * (vxA ^ 2) / ay,
ay / (vxA ^ 2) != 0,
sin(th) / cos(th) * xB != 0,
ay != 0)))
.SubstituteEqLs(vals)
.SimplifyEquation()
.AssertEqTo(
new Or(
new And(
xB == 89.312185996136435,
xB != 0),
new And(
xB == 7.0805039835788038E-15,
xB != 0)));
eqs
.SubstituteEqLs(zeros)
.EliminateVariables(vxB, vyB, d, xB, tAB)
.IsolateVariable(yB)
.LogicalExpand()
.CheckVariable(yB)
.AssertEqTo(
new And(
yB == 2 * (sin(th) ^ 2) * (vxA ^ 2) / ay / (cos(th) ^ 2),
-ay * (cos(th) ^ 2) / (sin(th) ^ 2) / (vxA ^ 2) / 2 != 0,
yB != 0,
ay != 0))
.SubstituteEqLs(vals)
.AssertEqTo(
new And(
yB == -62.537065888482395,
yB != 0));
eqs
.SubstituteEqLs(zeros)
.EliminateVariables(vxB, vyB, d, xB, yB)
.IsolateVariable(tAB)
.LogicalExpand().CheckVariable(ay).SimplifyEquation().SimplifyLogical()
.AssertEqTo(
new Or(
new And(
tAB == -(sin(th) * vxA / cos(th) + sqrt((sin(th) ^ 2) * (vxA ^ 2) / (cos(th) ^ 2))) / ay,
ay != 0,
sin(th) * tAB * vxA / cos(th) != 0),
new And(
tAB == -(sin(th) * vxA / cos(th) - sqrt((sin(th) ^ 2) * (vxA ^ 2) / (cos(th) ^ 2))) / ay,
ay != 0,
sin(th) * tAB * vxA / cos(th) != 0)))
.SubstituteEqLs(vals)
.CheckVariable(tAB).SimplifyEquation()
.AssertEqTo(
new And(
tAB == 3.5724874398454571,
tAB != 0));
eqs
.SubstituteEqLs(zeros)
.EliminateVariables(vxB, d, tAB, xB, yB)
.IsolateVariable(vyB)
.LogicalExpand()
.CheckVariable(ay)
.SimplifyEquation()
.CheckVariable(ay)
.AssertEqTo(
new Or(
new And(
vyB == -ay * (sin(th) * vxA / (ay * cos(th)) + sqrt((sin(th) ^ 2) * (vxA ^ 2) / ((ay ^ 2) * (cos(th) ^ 2)))),
sin(th) * vxA * vyB / (ay * cos(th)) != 0,
ay != 0),
new And(
vyB == -ay * (sin(th) * vxA / (ay * cos(th)) - sqrt((sin(th) ^ 2) * (vxA ^ 2) / ((ay ^ 2) * (cos(th) ^ 2)))),
sin(th) * vxA * vyB / (ay * cos(th)) != 0,
ay != 0)))
.SubstituteEqLs(vals)
.CheckVariable(vyB)
.SimplifyEquation()
.CheckVariable(vyB)
.AssertEqTo(
new And(
vyB == -35.010376910485483,
vyB != 0));
DoubleFloat.tolerance = null;
}
#endregion
#region PSE 5E P4.9
{
// In a local bar, a customer slides an empty beer mug
// down the counter for a refill. The bartender is momentarily
// distracted and does not see the mug, which slides
// off the counter and strikes the floor 1.40 m from the
// base of the counter. If the height of the counter is
// 0.860 m, (a) with what velocity did the mug leave the
// counter and (b) what was the direction of the mug’s
// velocity just before it hit the floor?
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
var xA = new Symbol("xA");
var yA = new Symbol("yA");
var xB = new Symbol("xB");
var yB = new Symbol("yB");
var vxA = new Symbol("vxA");
var vyA = new Symbol("vyA");
var vxB = new Symbol("vxB");
var vyB = new Symbol("vyB");
var tAB = new Symbol("tAB");
var ax = new Symbol("ax");
var ay = new Symbol("ay");
var thB = new Symbol("thB");
var vB = new Symbol("vB");
var eqs = new And(
vxB == vxA + ax * tAB,
vyB == vyA + ay * tAB,
tan(thB) == vyB / vxB,
xB == xA + vxA * tAB + ax * (tAB ^ 2) / 2,
yB == yA + vyA * tAB + ay * (tAB ^ 2) / 2,
xB != 0
);
var vals = new List<Equation>() { xA == 0, yA == 0.86, /* vxA */ vyA == 0, xB == 1.4, yB == 0, /* vxB vyB vB thB */ /* tAB */ ax == 0, ay == -9.8 };
var zeros = vals.Where(eq => eq.b == 0).ToList();
DoubleFloat.tolerance = 0.00001;
eqs
.SubstituteEqLs(zeros)
.EliminateVariables(thB, vxB, vyB, tAB)
.IsolateVariable(vxA)
.LogicalExpand()
.AssertEqTo(
new Or(
new And(
vxA == ay * (xB ^ 2) / yA / 4 * sqrt(-8 / ay * (xB ^ -2) * yA),
2 / ay * (xB ^ -2) * yA != 0,
xB != 0),
new And(
vxA == -ay * (xB ^ 2) / yA / 4 * sqrt(-8 / ay * (xB ^ -2) * yA),
2 / ay * (xB ^ -2) * yA != 0,
xB != 0)))
.SubstituteEqLs(vals)
.AssertEqTo(new Or(vxA == -3.3417722634053204, vxA == 3.3417722634053204));
eqs
.SubstituteEqLs(zeros)
.EliminateVariables(vxB, vyB, tAB, vxA)
.LogicalExpand()
.CheckVariable(xB)
.SimplifyLogical()
.IsolateVariable(thB)
.AssertEqTo(
new And(
-tan(thB) / ay != 0,
thB == new Atan(-2 * yA / xB),
xB != 0))
.SubstituteEqLs(vals)
.AssertEqTo(
new And(
0.1020408163265306 * tan(thB) != 0,
thB == -0.88760488150470185));
DoubleFloat.tolerance = null;
}
#endregion
#region SumDifferenceFormulaFunc
// sin(u) cos(v) - cos(u) sin(v) -> sin(u - v)
Func<MathObject, MathObject> SumDifferenceFormulaFunc = elt =>
{
if (elt is Sum)
{
var items = new List<MathObject>();
foreach (var item in (elt as Sum).elts)
{
if (
item is Product &&
(item as Product).elts[0] == -1 &&
(item as Product).elts[1] is Cos &&
(item as Product).elts[2] is Sin
)
{
var u_ = ((item as Product).elts[1] as Cos).args[0];
var v_ = ((item as Product).elts[2] as Sin).args[0];
Func<MathObject, bool> match = obj =>
obj is Product &&
(obj as Product).elts[0] is Cos &&
(obj as Product).elts[1] is Sin &&
((obj as Product).elts[1] as Sin).args[0] == u_ &&
((obj as Product).elts[0] as Cos).args[0] == v_;
if (items.Any(obj => match(obj)))
{
items = items.Where(obj => match(obj) == false).ToList();
items.Add(sin(u_ - v_));
}
else items.Add(item);
}
else items.Add(item);
}
return new Sum() { elts = items }.Simplify();
}
return elt;
};
#endregion
{
var u = new Symbol("u");
var v = new Symbol("v");
(sin(u) * cos(v) - cos(u) * sin(v))
.DeepSelect(SumDifferenceFormulaFunc)
.AssertEqTo(sin(u - v));
}
#region DoubleAngleFormulaFunc
// sin(u) cos(u) -> sin(2 u) / 2
Func<MathObject, MathObject> DoubleAngleFormulaFunc =
elt =>
{
if (elt is Product)
{
var items = new List<MathObject>();
foreach (var item in (elt as Product).elts)
{
if (item is Sin)
{
var sym = (item as Sin).args.First();
if (items.Any(obj => (obj is Cos) && (obj as Cos).args.First() == sym))
{
items = items.Where(obj => ((obj is Cos) && (obj as Cos).args.First() == sym) == false).ToList();
items.Add(sin(2 * sym) / 2);
}
else items.Add(item);
}
else if (item is Cos)
{
var sym = (item as Cos).args.First();
if (items.Any(obj => (obj is Sin) && (obj as Sin).args.First() == sym))
{
items = items.Where(obj => ((obj is Sin) && (obj as Sin).args.First() == sym) == false).ToList();
items.Add(sin(2 * sym) / 2);
}
else items.Add(item);
}
else items.Add(item);
}
return new Product() { elts = items }.Simplify();
}
return elt;
};
#endregion
#region PSE 5E P4.11
{
// One strategy in a snowball fight is to throw a first snowball
// at a high angle over level ground. While your opponent is watching
// the first one, you throw a second one at a low angle and timed
// to arrive at your opponent before or at the same time as the first one.
// Assume both snowballs are thrown with a speed of 25.0 m/s.
// The first one is thrown at an angle of 70.0° with respect to the horizontal.
// (a) At what angle should the second (lowangle)
// snowball be thrown if it is to land at the same
// point as the first?
// (b) How many seconds later should the second snowball
// be thrown if it is to land at the same time as the first?
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
var xA = new Symbol("xA");
var yA = new Symbol("yA");
var vxA = new Symbol("vxA");
var vyA = new Symbol("vyA");
var vA = new Symbol("vA");
var thA = new Symbol("thA");
var xB = new Symbol("xB");
var yB = new Symbol("yB");
var vxB = new Symbol("vxB");
var vyB = new Symbol("vyB");
var tAB = new Symbol("tAB");
var tAC = new Symbol("tAC");
var ax = new Symbol("ax");
var ay = new Symbol("ay");
var Pi = new Symbol("Pi");
var eqs = new And(
vxA == vA * cos(thA),
vyA == vA * sin(thA),
vxB == vxA + ax * tAB,
vyB == vyA + ay * tAB,
xB == xA + vxA * tAB + ax * (tAB ^ 2) / 2,
yB == yA + vyA * tAB + ay * (tAB ^ 2) / 2
);
DoubleFloat.tolerance = 0.00001;
{
var vals = new List<Equation>() { xA == 0, yA == 0, /* vxA vyA */ vA == 25.0, /* thA == 70.0, */ /* xB == 20.497, */ /* yB */ /* vxB */ vyB == 0, /* tAB */ ax == 0, ay == -9.8, Pi == Math.PI };
var zeros = vals.Where(eq => eq.b == 0).ToList();
{
// thA = ... || thA = ...
var expr = eqs
.SubstituteEqLs(zeros)
.EliminateVariables(yB, vxA, vyA, vxB, tAB)
.DeepSelect(DoubleAngleFormulaFunc)
.IsolateVariable(thA);
// th_delta = ...
var th1 = ((expr as Or).args[0] as Equation).b;
var th2 = ((expr as Or).args[1] as Equation).b;
var th_delta = new Symbol("th_delta");
eqs
.Add(th_delta == (th1 - th2).AlgebraicExpand())
.SubstituteEqLs(zeros)
.EliminateVariables(yB, vxA, vyA, vxB, tAB)
.DeepSelect(DoubleAngleFormulaFunc)
.EliminateVariable(xB)
.AssertEqTo(th_delta == asin(sin(2 * thA)) - Pi / 2)
.SubstituteEq(thA == (70).ToRadians())
.SubstituteEq(Pi == Math.PI)
.AssertEqTo(th_delta == -0.87266462599716454)
;
}
{
// tAB = ...
var tAB_eq = eqs
.SubstituteEqLs(zeros)
.EliminateVariables(yB, vxA, vyA, vxB, xB)
.IsolateVariable(tAB);
new And(
new Or(thA == (20).ToRadians(), thA == (70).ToRadians()),
tAB_eq,
tAC == tAB * 2)
.LogicalExpand()
.EliminateVariables(thA, tAB)
.AssertEqTo(new Or(
tAC == -2 * sin(Pi / 9) * vA / ay,
tAC == -2 * sin(7 * Pi / 18) * vA / ay))
.SubstituteEqLs(vals)
.AssertEqTo(
new Or(
tAC == 1.7450007312534115,
tAC == 4.794350106050552));
}
}
DoubleFloat.tolerance = null;
}
#endregion
#region PSE 5E P4.13
{
// An artillery shell is fired with an initial velocity of
// 300 m/s at 55.0° above the horizontal. It explodes on a
// mountainside 42.0 s after firing. What are the x and y
// coordinates of the shell where it explodes, relative to its
// firing point?
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
var xA = new Symbol("xA");
var yA = new Symbol("yA");
var vxA = new Symbol("vxA");
var vyA = new Symbol("vyA");
var vA = new Symbol("vA");
var thA = new Symbol("thA");
var xB = new Symbol("xB");
var yB = new Symbol("yB");
var vxB = new Symbol("vxB");
var vyB = new Symbol("vyB");
var tAB = new Symbol("tAB");
var ax = new Symbol("ax");
var ay = new Symbol("ay");
var Pi = new Symbol("Pi");
var eqs = new And(
vxA == vA * cos(thA),
vyA == vA * sin(thA),
vxB == vxA + ax * tAB,
vyB == vyA + ay * tAB,
xB == xA + vxA * tAB + ax * (tAB ^ 2) / 2,
yB == yA + vyA * tAB + ay * (tAB ^ 2) / 2
);
DoubleFloat.tolerance = 0.00001;
{
var vals = new List<Equation>() { xA == 0, yA == 0, /* vxA vyA */ vA == 300.0, thA == (55).ToRadians(), /* xB yB vxB vyB */ tAB == 42, ax == 0, ay == -9.8, Pi == Math.PI };
var zeros = vals.Where(eq => eq.b == 0).ToList();
{
eqs
.SubstituteEqLs(zeros)
.EliminateVariable(vxA)
.EliminateVariable(vyA)
.AssertEqTo(
new And(
vxB == cos(thA) * vA,
vyB == ay * tAB + sin(thA) * vA,
xB == cos(thA) * tAB * vA,
yB == ay * (tAB ^ 2) / 2 + sin(thA) * tAB * vA))
.SubstituteEqLs(vals)
.AssertEqTo(
new And(
vxB == 172.07293090531385,
vyB == -165.85438671330249,
xB == 7227.0630980231817,
yB == 1677.7157580412968))
;
}
}
DoubleFloat.tolerance = null;
}
#endregion
#region PSE 5E P4.15
{
// A projectile is fired in such a way that its horizontal
// range is equal to three times its maximum height.
//
// What is the angle of projection?
// Give your answer to three significant figures.
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
var xA = new Symbol("xA");
var yA = new Symbol("yA");
var vxA = new Symbol("vxA");
var vyA = new Symbol("vyA");
var vA = new Symbol("vA");
var thA = new Symbol("thA");
var xB = new Symbol("xB");
var yB = new Symbol("yB");
var vxB = new Symbol("vxB");
var vyB = new Symbol("vyB");
var xC = new Symbol("xC");
var yC = new Symbol("yC");
var vxC = new Symbol("vxC");
var vyC = new Symbol("vyC");
var tAB = new Symbol("tAB");
var tBC = new Symbol("tBC");
var ax = new Symbol("ax");
var ay = new Symbol("ay");
var Pi = new Symbol("Pi");
var eqs = new And(
xC - xA == 3 * yB,
tAB == tBC,
vxA == vA * cos(thA),
vyA == vA * sin(thA),
vxB == vxA + ax * tAB,
vyB == vyA + ay * tAB,
xB == xA + vxA * tAB + ax * (tAB ^ 2) / 2,
yB == yA + vyA * tAB + ay * (tAB ^ 2) / 2,
vxC == vxB + ax * tBC,
vyC == vyB + ay * tBC,
xC == xB + vxB * tBC + ax * (tBC ^ 2) / 2,
yC == yB + vyB * tBC + ay * (tBC ^ 2) / 2
);
DoubleFloat.tolerance = 0.00001;
{
var vals = new List<Equation>()
{
xA == 0, yA == 0, /* vxA vyA vA thA */ /* xB yB vxB */ vyB == 0, /* tAB tBC */
/* xC */ yC == 0,
ax == 0, ay == -9.8, Pi == Math.PI
};
var zeros = vals.Where(eq => eq.b == 0).ToList();
eqs
.SubstituteEqLs(zeros)
.EliminateVariables(xC, tAB, vxA, vyA, vxB, xB, yB, vxC, vyC, tBC)
.IsolateVariable(thA)
.AssertEqTo(thA == new Atan(new Integer(4) / 3));
}
DoubleFloat.tolerance = null;
}
#endregion
#region PSE 5E P4.17
{
// A cannon with a muzzle speed of 1000 m/s is used to
// start an avalanche on a mountain slope. The target is
// 2000 m from the cannon horizontally and 800 m above
// the cannon.
//
// At what angle, above the horizontal, should the cannon be fired?
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
var xA = new Symbol("xA");
var yA = new Symbol("yA");
var vxA = new Symbol("vxA");
var vyA = new Symbol("vyA");
var vA = new Symbol("vA");
var thA = new Symbol("thA");
var xB = new Symbol("xB");
var yB = new Symbol("yB");
var vxB = new Symbol("vxB");
var vyB = new Symbol("vyB");
var xC = new Symbol("xC");
var yC = new Symbol("yC");
var vxC = new Symbol("vxC");
var vyC = new Symbol("vyC");
var tAB = new Symbol("tAB");
var tBC = new Symbol("tBC");
var ax = new Symbol("ax");
var ay = new Symbol("ay");
var Pi = new Symbol("Pi");
var phi = new Symbol("phi");
var eqs = new And(
vxA == vA * cos(thA),
vyA == vA * sin(thA),
vxB == vxA + ax * tAB,
vyB == vyA + ay * tAB,
xB == xA + vxA * tAB + ax * (tAB ^ 2) / 2,
yB == yA + vyA * tAB + ay * (tAB ^ 2) / 2
);
DoubleFloat.tolerance = 0.00001;
{
var vals = new List<Equation>()
{
xA == 0, yA == 0, /* vxA vyA */ vA == 1000, /* thA */
xB == 2000, yB == 800.0, /* vxB vyB */
/* tAB */ ax == 0, ay == -9.8, Pi == Math.PI
};
var zeros = vals.Where(eq => eq.b == 0).ToList();
{
eqs
.SubstituteEqLs(zeros)
.EliminateVariables(vxA, vyA, vxB, vyB, tAB)
.MultiplyBothSidesBy(cos(thA) ^ 2).AlgebraicExpand()
.Substitute(cos(thA) ^ 2, (1 + cos(2 * thA)) / 2)
.DeepSelect(DoubleAngleFormulaFunc).AlgebraicExpand()
.AddToBothSides(-sin(2 * thA) * xB / 2)
.AddToBothSides(-yB / 2)
.MultiplyBothSidesBy(2 / xB).AlgebraicExpand()
// yB / xB = tan(phi)
// yB / xB = sin(phi) / cos(phi)
// phi = atan(yB / xB)
.Substitute(cos(2 * thA) * yB / xB, cos(2 * thA) * sin(phi) / cos(phi))
.MultiplyBothSidesBy(cos(phi)).AlgebraicExpand()
.DeepSelect(SumDifferenceFormulaFunc)
.IsolateVariable(thA)
.Substitute(phi, new Atan(yB / xB).Simplify())
.AssertEqTo(
new Or(
thA == -(asin(ay * cos(atan(yB / xB)) * (vA ^ -2) * xB + -1 * cos(atan(yB / xB)) * yB / xB) - atan(yB / xB)) / 2,
thA == -(-asin(ay * cos(atan(yB / xB)) * (vA ^ -2) * xB - cos(atan(yB / xB)) * yB / xB) - atan(yB / xB) + Pi) / 2))
.SubstituteEqLs(vals)
.AssertEqTo(
new Or(
thA == 0.39034573609628065,
thA == -1.5806356857788124))
;
}
}
DoubleFloat.tolerance = null;
}
#endregion
#endregion
#region PSE 5E Example 4.3
{
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA =
new Obj()
{
position = new Point(0, 0),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = new Integer(0)
};
var objB =
new Obj()
{
velocity = new Point(objA.velocity.x, 0),
acceleration = _g
};
var timeB = Calc.Time(objA, objB);
var timeC = timeB * 2;
objB = objA.AtTime(timeB);
var objC = objA.AtTime(timeC);
//Console.WriteLine("How far does he dump in the horizontal direction?");
AssertIsTrue(objC.position.x == 2 * Trig.Cos(thA) * Trig.Sin(thA) * (vA ^ 2) / g);
//Console.WriteLine("What is the maximum height reached?");
AssertIsTrue(objB.position.y == (Trig.Sin(thA) ^ 2) * (vA ^ 2) / 2 / g);
// Console.WriteLine("Distance jumped: ");
var deg = 3.14159 / 180.0;
AssertIsTrue(
objC.position.x
// .Substitute(thA, Trig.ToRadians(20))
.Substitute(thA, 20 * deg)
.Substitute(g, 9.8)
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 11)
==
7.9364536850196412);
//Console.WriteLine("Maximum height reached: ");
AssertIsTrue(
objB.position.y
.Substitute(g, 9.8)
.Substitute(thA, Trig.ToRadians(20))
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 11)
==
0.72215756424454336);
}
#endregion
#region PSE 5E EXAMPLE 4.5
{
// A stone is thrown from the top of a building upward at an
// angle of 30.0° to the horizontal and with an initial speed of
// 20.0 m/s, as shown in Figure 4.12. If the height of the building
// is 45.0 m, (a) how long is it before the stone hits the ground?
// (b) What is the speed of the stone just before it strikes the
// ground?
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA = new Obj()
{
position = new Point(0, 0),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = new Integer(0)
};
var objB = new Obj()
{
velocity = new Point(objA.velocity.x, 0),
acceleration = _g,
};
var timeB = Calc.Time(objA, objB);
objB = objA.AtTime(timeB);
var timeC = timeB * 2;
var objC = objA.AtTime(timeC);
var yD = new Symbol("yD");
var objD = new Obj()
{
position = new Point(null, yD),
velocity = new Point(objA.velocity.x, null),
acceleration = _g
};
var timeAD = Calc.Time(objA, objD, 1);
objD = objA.AtTime(timeAD);
// "How long is it before the stone hits the ground?".Disp();
// "Symbolic answer:".Disp();
AssertIsTrue(
timeAD
==
-1 * (g ^ -1) * (-1 * Trig.Sin(thA) * vA + -1 * (((Trig.Sin(thA) ^ 2) * (vA ^ 2) + -2 * g * yD) ^ (new Integer(1) / 2))));
// "Numeric answer:".Disp();
AssertEqual(
(DoubleFloat)
timeAD
.Substitute(g, 9.8)
.Substitute(thA, (30).ToRadians())
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 20)
.Substitute(yD, -45),
new DoubleFloat(4.21804787012706),
0.0001);
// "What is the speed of the stone just before it strikes the ground?".Disp();
// "Symbolic answer:".Disp();
AssertIsTrue(
objD.velocity.Norm()
==
(((Trig.Cos(thA) ^ 2) * (vA ^ 2) + (Trig.Sin(thA) ^ 2) * (vA ^ 2) + -2 * g * yD) ^ (new Integer(1) / 2)));
// "Numeric answer:".Disp();
AssertEqual(
(DoubleFloat)
objD.velocity.Norm()
.Substitute(g, 9.8)
.Substitute(thA, (30).ToRadians())
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 20)
.Substitute(yD, -45),
new DoubleFloat(35.805027579936315),
0.1);
}
#endregion
#region PSE 5E EXAMPLE 4.6
{
// An Alaskan rescue plane drops a package of emergency rations
// to a stranded party of explorers, as shown in Figure
// 4.13. If the plane is traveling horizontally at 40.0 m/s and is
// 100 m above the ground, where does the package strike the
// ground relative to the point at which it was released?
var xA = new Symbol("xA"); // position.x at point A
var yA = new Symbol("yA"); // position.y at point A
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA = new Obj() // obj at the initial position
{
position = new Point(xA, yA),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = 0
};
var objB = new Obj() // obj at the final position
{
position = new Point(null, 0),
velocity = new Point(objA.velocity.x, null),
acceleration = _g
};
var timeB = Calc.Time(objA, objB, 1);
objB = objA.AtTime(timeB);
//"Where does the package strike the ground relative to the point at which it was released?".Disp(); "".Disp();
//"symbolic:".Disp();
//objB.position.x.Disp(); "".Disp();
AssertIsTrue(
objB.position.x
==
xA - cos(thA) / g * vA * (-sin(thA) * vA - (((sin(thA) ^ 2) * (vA ^ 2) + 2 * g * yA) ^ new Integer(1) / 2)));
//"numeric:".Disp();
//objB.position.x
// .Substitute(xA, 0)
// .Substitute(yA, 100)
// .Substitute(vA, 40)
// .Substitute(thA, 0.0)
// .Substitute(g, 9.8)
// .Disp();
AssertEqual(
objB.position.x
.Substitute(xA, 0)
.Substitute(yA, 100)
.Substitute(vA, 40)
.Substitute(thA, 0.0)
.Substitute(g, 9.8),
180.70158058105025);
//"".Disp();
//("What are the horizontal and vertical components " +
// "of the velocity of the package just before it hits the ground?").Disp(); "".Disp();
//"symbolic velocity.x:".Disp();
//objB.velocity.x.Disp(); "".Disp();
AssertIsTrue(objB.velocity.x == cos(thA) * vA);
//"symbolic velocity.y:".Disp();
//objB.velocity.y.Disp(); "".Disp();
AssertIsTrue(
objB.velocity.y
==
-1 * (((sin(thA) ^ 2) * (vA ^ 2) + 2 * g * yA) ^ (new Integer(1) / 2)));
//"numeric velocity.x:".Disp();
//objB.velocity.x
// .Substitute(xA, 0)
// .Substitute(yA, 100)
// .Substitute(vA, 40)
// .Substitute(thA, 0.0)
// .Substitute(g, 9.8)
// .Disp(); "".Disp();
AssertEqual(
objB.velocity.x
.Substitute(xA, 0)
.Substitute(yA, 100)
.Substitute(vA, 40)
.Substitute(thA, 0.0)
.Substitute(g, 9.8),
40);
//"numeric velocity.y:".Disp();
//objB.velocity.y
// .Substitute(xA, 0)
// .Substitute(yA, 100)
// .Substitute(vA, 40)
// .Substitute(thA, 0.0)
// .Substitute(g, 9.8)
// .Disp(); "".Disp();
AssertEqual(
objB.velocity.y
.Substitute(xA, 0)
.Substitute(yA, 100)
.Substitute(vA, 40)
.Substitute(thA, 0.0)
.Substitute(g, 9.8),
-44.271887242357316);
}
#endregion
#region PSE 5E EXAMPLE 4.7
{
// A ski jumper leaves the ski track moving in the horizontal
// direction with a speed of 25.0 m/s, as shown in Figure 4.14.
// The landing incline below him falls off with a slope of 35.0°.
// Where does he land on the incline?
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var th = new Symbol("th"); // angle of incline
var objA = new Obj()
{
position = new Point(0, 0),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = 0
};
Func<MathObject, MathObject> numeric = obj =>
obj
.Substitute(vA, 25)
.Substitute(thA, 0.0)
.Substitute(th, (-35).ToRadians())
.Substitute(Trig.Pi, 3.14159)
.Substitute(g, 9.8);
var intersection = objA.ProjectileInclineIntersection(th);
Action nl = () => "".Disp();
// "Where does he land on the incline?".Disp(); nl();
// "x position (symbolic):".Disp();
// intersection.x.Disp(); nl();
AssertIsTrue(
intersection.x
==
-2 * (cos(th) ^ -1) * (cos(thA) ^ 2) * (g ^ -1) * (sin(th) + -1 * cos(th) * (cos(thA) ^ -1) * sin(thA)) * (vA ^ 2));
//"y position (symbolic):".Disp();
//intersection.y.Disp(); nl();
AssertIsTrue(
intersection.y
==
-2 * (cos(th) ^ -2) * (cos(thA) ^ 2) / g * sin(th) * (sin(th) + -1 * cos(th) * (cos(thA) ^ -1) * sin(thA)) * (vA ^ 2));
//"x position (numeric):".Disp();
//numeric(intersection.x).Disp(); nl();
AssertEqual(numeric(intersection.x), 89.3120879153208);
//"y position (numeric):".Disp();
//numeric(intersection.y).Disp(); nl();
AssertEqual(numeric(intersection.y), -62.536928534704884);
var objB = new Obj()
{
position = intersection,
acceleration = _g
};
//"Determine how long the jumper is airborne".Disp(); nl();
//"symbolic:".Disp();
var timeB = Calc.Time(objA, objB, 1);
// timeB.Disp(); nl();
Func<MathObject, MathObject> sqrt = obj => obj ^ (new Integer(1) / 2);
AssertIsTrue(
timeB
==
-1 / g *
(-sin(thA) * vA -
sqrt(
(sin(thA) ^ 2) * (vA ^ 2) + 4 * (cos(th) ^ -2) * (cos(thA) ^ 2) * sin(th) *
(sin(th) - cos(th) / cos(thA) * sin(thA)) *
(vA ^ 2))));
//"numeric:".Disp();
//numeric(timeB).Disp(); nl();
AssertEqual(numeric(timeB), 3.5724835166128317);
objB = objA.AtTime(timeB);
//"Determine his vertical component of velocity just before he lands".Disp();
//nl();
//"symbolic:".Disp();
//objB.velocity.y.Disp(); nl();
AssertIsTrue(
objB.velocity.y
==
-sqrt(
(sin(thA) ^ 2) * (vA ^ 2)
+
4 * (cos(th) ^ -2) * (cos(thA) ^ 2) * sin(th) *
(sin(th) - cos(th) * (cos(thA) ^ -1) * sin(thA)) *
(vA ^ 2)));
//"numeric:".Disp();
//numeric(objB.velocity.y).Disp();
AssertEqual(
numeric(objB.velocity.y),
-35.010338462805755);
}
#endregion
#region PSE 5E PROBLEM 4.11
{
// One strategy in a snowball fight is to throw a first snowball at a
// high angle over level ground. While your opponent is watching the
// first one, you throw a second one at a low angle and timed to arrive
// at your opponent before or at the same time as the first one. Assume
// both snowballs are thrown with a speed of 25.0 m/s. The first one is
// thrown at an angle of 70.0° with respect to the horizontal.
//
// (a) At what angle should the second (low-angle) snowball be thrown
// if it is to land at the same point as the first?
//
// (b) How many seconds later should the second snowball be thrown if it
// is to land at the same time as the first?
var xA = new Symbol("xA"); // position.x at point A
var yA = new Symbol("yA"); // position.y at point A
var th1A = new Symbol("th1A"); // angle of snowball 1 at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
//Func<MathObject, MathObject> numeric = obj =>
// obj
// .Substitute(xA, 0)
// .Substitute(xB, 1.4)
// .Substitute(yA, 0.86)
// .Substitute(g, 9.8)
// .Substitute(Trig.Pi, 3.14159);
var obj1A = new Obj() // snowball 1 at initial point
{
position = new Point(xA, yA),
velocity = Point.FromAngle(th1A, vA),
acceleration = _g,
time = 0
};
var obj1B = new Obj() // snowball 1 at final point
{
position = new Point(null, 0),
velocity = new Point(obj1A.velocity.x, null),
acceleration = _g
};
var time1B = Calc.Time(obj1A, obj1B, 1);
obj1B = obj1A.AtTime(time1B);
var obj2A = new Obj() // snowball 2 at initial point
{
position = obj1A.position,
speed = vA,
acceleration = _g
};
var obj2B = new Obj() // snowball 2 at final point
{
position = obj1B.position,
acceleration = _g
};
//Calc.InitialAngle(obj2A, obj2B, 1, 0)
// .Substitute(yA, 0)
// .Substitute(th1A, (70).ToRadians())
// .Substitute(vA, 25)
// .Substitute(Trig.Pi, 3.14159)
// .Substitute(g, 9.8)
// .ToDegrees()
// .Substitute(Trig.Pi, 3.14159)
// .Disp();
var th2 = Calc.InitialAngle(obj2A, obj2B, 0, 0);
//("At what angle should the second (low-angle) snowball " +
//"be thrown if it is to land at the same point as the first?").Disp();
//"".Disp();
//"symbolic:".Disp();
//th2.Disp(); "".Disp();
//"numeric:".Disp();
AssertEqual(
th2
.ToDegrees()
.Substitute(yA, 0)
.Substitute(th1A, (70).ToRadians())
.Substitute(vA, 25)
.Substitute(g, 9.8)
.Substitute(Trig.Pi, Math.PI),
20.000000000000007);
//"".Disp();
obj2A.velocity = Point.FromAngle(th2, vA);
var time2B = Calc.Time(obj2A, obj2B, 1);
//("How many seconds later should the second snowball be thrown if it " +
//"is to land at the same time as the first?").Disp();
//"".Disp();
//"symbolic:".Disp();
//(time1B - time2B).Disp(); "".Disp();
//"numeric:".Disp();
//(time1B - time2B)
// .Substitute(yA, 0)
// .Substitute(th1A, (70).ToRadians())
// .Substitute(vA, 25)
// .Substitute(Trig.Pi, 3.14159)
// .Substitute(g, 9.8)
// .Disp();
AssertEqual(
(time1B - time2B)
.Substitute(yA, 0)
.Substitute(th1A, (70).ToRadians())
.Substitute(vA, 25)
.Substitute(Trig.Pi, 3.14159)
.Substitute(g, 9.8),
3.0493426265020469);
//Console.ReadLine();
}
#endregion
#region PSE 5E PROBLEM 4.17
{
// A cannon with a muzzle speed of 1 000 m/s is used to
// start an avalanche on a mountain slope. The target is
// 2 000 m from the cannon horizontally and 800 m above
// the cannon. At what angle, above the horizontal, should
// the cannon be fired?
var xA = new Symbol("xA"); // position.x at point A
var yA = new Symbol("yA"); // position.y at point A
var thA = new Symbol("thA"); // angle of snowball 1 at point A
var vA = new Symbol("vA"); // velocity at point A
var xB = new Symbol("xB"); // position.x at point A
var yB = new Symbol("yB"); // position.y at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA = new Obj()
{
position = new Point(xA, yA),
speed = vA,
acceleration = _g,
time = 0
};
var objB = new Obj()
{
position = new Point(xB, yB),
acceleration = _g
};
//"At what angle, above the horizontal, should the cannon be fired?".Disp();
AssertEqual(
Calc.InitialAngle(objA, objB)
.ToDegrees()
.Substitute(xA, 0)
.Substitute(yA, 0)
.Substitute(xB, 2000.0)
.Substitute(yB, 800)
.Substitute(vA, 1000)
.Substitute(g, 9.8)
.Substitute(Trig.Pi, Math.PI),
22.365163229244317);
//Calc.InitialAngle(objA, objB)
// .ToDegrees()
// .Substitute(xA, 0)
// .Substitute(yA, 0)
// .Substitute(xB, 2000.0)
// .Substitute(yB, 800)
// .Substitute(vA, 1000)
// .Substitute(g, 9.8)
// .Substitute(Trig.Pi, Math.PI)
// .Disp();
}
#endregion
#region PSE 5E PROBLEM 4.24
{
// A bag of cement of weight 325 N hangs from three
// wires as shown in Figure P5.24. Two of the wires make
// angles th1 = 60.0° and th2 = 25.0° with the horizontal. If
// the system is in equilibrium, find the tensions
// T1, T2, and T3 in the wires.
var F1 = new Symbol("F1");
var F2 = new Symbol("F2");
var F3 = new Symbol("F3");
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");
var th3 = new Symbol("th3");
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2 };
var _F3 = new Point() { magnitude = F3, angle = th3 };
var m = new Symbol("m");
var obj = new Obj();
obj.acceleration.x = 0;
obj.acceleration.y = 0;
obj.mass = m;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
obj.forces.Add(_F3);
//"F1 magnitude, symbolic:".Disp(); "".Disp();
//obj.ForceMagnitude(_F1).Disp(); "".Disp();
//"F1 magnitude, numeric:".Disp(); "".Disp();
//obj.ForceMagnitude(_F1)
// .Substitute(F3, 325)
// .Substitute(th1, (180 - 60).ToRadians())
// .Substitute(th2, (25).ToRadians())
// .Substitute(th3, (270).ToRadians())
// .Substitute(Trig.Pi, Math.PI)
// .Disp();
AssertEqual(
obj.ForceMagnitude(_F1)
.Substitute(F3, 325)
.Substitute(th1, (180 - 60).ToRadians())
.Substitute(th2, (25).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI),
295.67516405290525);
// "".Disp();
//"F3 magnitude, numeric:".Disp(); "".Disp();
//obj.ForceMagnitude(_F2)
// .Substitute(F3, 325)
// .Substitute(th1, (180 - 60).ToRadians())
// .Substitute(th2, (25).ToRadians())
// .Substitute(th3, (270).ToRadians())
// .Substitute(Trig.Pi, Math.PI)
// .Disp();
AssertEqual(
obj.ForceMagnitude(_F2)
.Substitute(F3, 325)
.Substitute(th1, (180 - 60).ToRadians())
.Substitute(th2, (25).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI),
163.12072360079395);
}
#endregion
#region PSE 5E PROBLEM 5.26
{
// You are a judge in a children’s kite-flying contest, and
// two children will win prizes for the kites that pull most
// strongly and least strongly on their strings. To measure
// string tensions, you borrow a weight hanger, some slotted
// weights, and a protractor from your physics teacher
// and use the following protocol, illustrated in Figure
// P5.26: Wait for a child to get her kite well-controlled,
// hook the hanger onto the kite string about 30 cm from
// her hand, pile on weights until that section of string is
// horizontal, record the mass required, and record the
// angle between the horizontal and the string running up
// to the kite. (a) Explain how this method works. As you
// construct your explanation, imagine that the children’s
// parents ask you about your method, that they might
// make false assumptions about your ability without concrete
// evidence, and that your explanation is an opportunity to
// give them confidence in your evaluation tech-nique.
// (b) Find the string tension if the mass required
// to make the string horizontal is 132 g and the angle of
// the kite string is 46.3°.
var F1 = new Symbol("F1");
var F2 = new Symbol("F2");
var F3 = new Symbol("F3");
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");
var th3 = new Symbol("th3");
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2 };
var _F3 = new Point() { magnitude = F3, angle = th3 };
var m = new Symbol("m");
var obj = new Obj();
obj.acceleration.x = 0;
obj.acceleration.y = 0;
obj.mass = m;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
obj.forces.Add(_F3);
//"Tension in line to kite:".Disp(); "".Disp();
//obj.ForceMagnitude(_F2)
// .Substitute(th1, (180).ToRadians())
// .Substitute(th2, (46.3 * Math.PI / 180))
// .Substitute(th3, (270).ToRadians())
// .Substitute(F3, 0.132 * 9.8)
// .Substitute(Trig.Pi, Math.PI)
// .Disp();
AssertEqual(
obj.ForceMagnitude(_F2)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (46.3 * Math.PI / 180))
.Substitute(th3, (270).ToRadians())
.Substitute(F3, 0.132 * 9.8)
.Substitute(Trig.Pi, Math.PI),
1.7892929261294661);
//Console.ReadLine();
}
#endregion
#region PSE 5E PROBLEM 5.28
{
// A fire helicopter carries a 620-kg bucket of water at the
// end of a cable 20.0 m long. As the aircraft flies back
// from a fire at a constant speed of 40.0 m/s, the cable
// makes an angle of 40.0° with respect to the vertical.
// (a) Determine the force of air resistance on the bucket.
// (b) After filling the bucket with sea water, the pilot re-
// turns to the fire at the same speed with the bucket now
// making an angle of 7.00° with the vertical. What is the
// mass of the water in the bucket?
var F1 = new Symbol("F1"); // force of air resistance
var F2 = new Symbol("F2"); // force of cable
var F3 = new Symbol("F3"); // force of gravity
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");
var th3 = new Symbol("th3");
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2 };
var _F3 = new Point() { magnitude = F3, angle = th3 };
var m = new Symbol("m");
var obj = new Obj();
obj.acceleration.x = 0;
obj.acceleration.y = 0;
obj.mass = m;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
obj.forces.Add(_F3);
//"Force of air resistance on the bucket:".Disp(); "".Disp();
var FAir =
obj.ForceMagnitude(_F1)
.Substitute(F3, 620 * 9.8)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (90 - 40).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI);
AssertEqual(
obj.ForceMagnitude(_F1)
.Substitute(F3, 620 * 9.8)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (90 - 40).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI),
5098.3693590331513);
//"".Disp();
_F1.magnitude = FAir;
_F3.magnitude = null;
var FBucketWithWater =
obj.ForceMagnitude(_F3)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (90 - 7).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI);
//"What is the mass of the water in the bucket?".Disp(); "".Disp();
//(FBucketWithWater / 9.8 - 620).Disp();
AssertEqual(
(FBucketWithWater / 9.8 - 620),
3617.0292120139611);
}
#endregion
#region PSE 5E PROBLEM 5.30
{
// A simple accelerometer is constructed by suspending a
// mass m from a string of length L that is tied to the top
// of a cart. As the cart is accelerated the string-mass
// system makes a constant angle th with the vertical.
// (a) Assuming that the string mass is negligible compared
// with m, derive an expression for the cart’s acceleration
// in terms of and show that it is independent of
// the mass mand the length L.
// (b) Determine the acceleration of the cart when th = 23.0°.
var F1 = new Symbol("F1"); // force of string
var F2 = new Symbol("F2"); // force of gravity
var th1 = new Symbol("th1");
var th2 = new Symbol("th2"); ;
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2, magnitude = F2 };
var m = new Symbol("m");
var g = new Symbol("g");
var obj = new Obj() { mass = m };
obj.acceleration.y = 0;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
_F1.magnitude = obj.ForceMagnitude(_F1);
//("Derive an expression for the cart’s acceleration in terms " +
//"of and show that it is independent of the mass mand the length L:").Disp();
//"".Disp();
//obj.AccelerationX()
// .Substitute(F2, m * g)
// .Substitute(Trig.Cos(th2), 0)
// .Substitute(Trig.Sin(th2), -1)
// .Disp();
//"".Disp();
//"Determine the acceleration of the cart when th = 23.0°".Disp(); "".Disp();
//obj.AccelerationX()
// .Substitute(F2, m * g)
// .Substitute(Trig.Cos(th2), 0)
// .Substitute(Trig.Sin(th2), -1)
// .Substitute(th1, (90 - 23).ToRadians())
// .Substitute(Trig.Pi, Math.PI)
// .Substitute(g, 9.8)
// .Disp();
AssertEqual(
obj.AccelerationX()
.Substitute(F2, m * g)
.Substitute(Trig.Cos(th2), 0)
.Substitute(Trig.Sin(th2), -1)
.Substitute(th1, (90 - 23).ToRadians())
.Substitute(Trig.Pi, Math.PI)
.Substitute(g, 9.8),
4.1598531988541261);
}
#endregion
#region PSE 5E PROBLEM 5.31
{
// Two people pull as hard as they can on ropes attached
// to a boat that has a mass of 200 kg. If they pull in the
// same direction, the boat has an acceleration of
// 1.52 m/s^2 to the right. If they pull in opposite directions,
// the boat has an acceleration of 0.518 m/s^2
// to the left. What is the force exerted by each person on the
// boat? (Disregard any other forces on the boat.)
// Trig.Cos(new DoubleFloat(Math.PI)).Disp();
var m = new Symbol("m");
var aAx = new Symbol("aAx");
var aBx = new Symbol("aBx");
var objA = new Obj() { mass = m };
objA.acceleration.x = aAx;
var _F1A = new Point() { angle = 0 };
var _F2A = new Point() { angle = 0 };
objA.forces.Add(_F1A);
objA.forces.Add(_F2A);
var objB = new Obj() { mass = m };
objB.acceleration.x = aBx;
var _F1B = new Point() { angle = 0 };
var _F2B = new Point() { angle = new DoubleFloat(Math.PI) };
objB.forces.Add(_F1B);
objB.forces.Add(_F2B);
//"force 1 magnitude (symbolic):".Disp(); "".Disp();
//Calc.ForceMagnitude(objA, objB, _F1A, _F1B)
// .Substitute(Trig.Cos(0), 1)
// .Disp();
//"force 1 magnitude (numeric):".Disp(); "".Disp();
//Calc.ForceMagnitude(objA, objB, _F1A, _F1B)
// .Substitute(Trig.Cos(0), 1)
// .Substitute(m, 200)
// .Substitute(aAx, 1.52)
// .Substitute(aBx, -0.518)
// .Disp();
AssertEqual(
Calc.ForceMagnitude(objA, objB, _F1A, _F1B)
.Substitute(Trig.Cos(0), 1)
.Substitute(m, 200)
.Substitute(aAx, 1.52)
.Substitute(aBx, -0.518),
100.19999999999999);
//"".Disp();
//"force 2 magnitude (symbolic):".Disp(); "".Disp();
//Calc.ForceMagnitude(objA, objB, _F2A, _F2B)
// .Substitute(Trig.Cos(0), 1)
// .Disp();
//"force 2 magnitude (numeric):".Disp(); "".Disp();
//Calc.ForceMagnitude(objA, objB, _F2A, _F2B)
// .Substitute(Trig.Cos(0), 1)
// .Substitute(m, 200)
// .Substitute(aAx, 1.52)
// .Substitute(aBx, -0.518)
// .Disp();
//"".Disp();
AssertEqual(
Calc.ForceMagnitude(objA, objB, _F2A, _F2B)
.Substitute(Trig.Cos(0), 1)
.Substitute(m, 200)
.Substitute(aAx, 1.52)
.Substitute(aBx, -0.518),
203.8);
}
#endregion
Console.WriteLine("Testing complete");
Console.ReadLine();
}
static void Main(string[] args)
{
// Two people pull as hard as they can on ropes attached
// to a boat that has a mass of 200 kg. If they pull in the
// same direction, the boat has an acceleration of
// 1.52 m/s^2 to the right. If they pull in opposite directions,
// the boat has an acceleration of 0.518 m/s^2
// to the left. What is the force exerted by each person on the
// boat? (Disregard any other forces on the boat.)
// Trig.Cos(new DoubleFloat(Math.PI)).Disp();
var m = new Symbol("m");
var aAx = new Symbol("aAx");
var aBx = new Symbol("aBx");
var objA = new Obj() { mass = m };
objA.acceleration.x = aAx;
var _F1A = new Point() { angle = 0 };
var _F2A = new Point() { angle = 0 };
objA.forces.Add(_F1A);
objA.forces.Add(_F2A);
var objB = new Obj() { mass = m };
objB.acceleration.x = aBx;
var _F1B = new Point() { angle = 0 };
var _F2B = new Point() { angle = new DoubleFloat(Math.PI) };
objB.forces.Add(_F1B);
objB.forces.Add(_F2B);
"force 1 magnitude (symbolic):".Disp(); "".Disp();
Calc.ForceMagnitude(objA, objB, _F1A, _F1B)
.Substitute(Trig.Cos(0), 1)
.Disp();
"force 1 magnitude (numeric):".Disp(); "".Disp();
Calc.ForceMagnitude(objA, objB, _F1A, _F1B)
.Substitute(Trig.Cos(0), 1)
.Substitute(m, 200)
.Substitute(aAx, 1.52)
.Substitute(aBx, -0.518)
.Disp();
"".Disp();
"force 2 magnitude (symbolic):".Disp(); "".Disp();
Calc.ForceMagnitude(objA, objB, _F2A, _F2B)
.Substitute(Trig.Cos(0), 1)
.Disp();
"force 2 magnitude (numeric):".Disp(); "".Disp();
Calc.ForceMagnitude(objA, objB, _F2A, _F2B)
.Substitute(Trig.Cos(0), 1)
.Substitute(m, 200)
.Substitute(aAx, 1.52)
.Substitute(aBx, -0.518)
.Disp();
"".Disp();
Console.ReadLine();
}
public static MathObject Time(Obj a, Obj b, int solution = 0)
{
if (a.velocity.x != null &&
b.velocity.x != null &&
a.acceleration.x != null &&
a.acceleration.x != new DoubleFloat(0.0) &&
a.acceleration.x != new Integer(0))
return (b.velocity.x - a.velocity.x) / a.acceleration.x;
if (a.velocity.y != null &&
b.velocity.y != null &&
a.acceleration.y != null &&
a.acceleration.y != new DoubleFloat(0.0) &&
a.acceleration.y != new Integer(0))
return (b.velocity.y - a.velocity.y) / a.acceleration.y;
// yf = yi + vyi * t + 1/2 * ay * t^2
// 0 = 1/2 * ay * t^2 + vyi * t + yi - yf
// apply quadratic equation to find t
if (a.position.y != null &&
b.position.y != null &&
a.velocity.y != null &&
a.acceleration.y != null &&
a.acceleration.y != new Integer(0) &&
a.acceleration.y != new DoubleFloat(0.0))
{
var half = new Integer(1) / 2;
return
Misc.QuadraticEquation(
half * a.acceleration.y,
a.velocity.y,
a.position.y - b.position.y,
solution);
}
throw new Exception();
}
public static Point InitialVelocity(Obj a, Obj b)
{
if (a.time != null && b.time != null)
{
var dt = b.time - a.time;
if (Misc.NotNull(
a.position.x,
a.position.y,
b.position.x,
b.position.y,
a.acceleration.x,
a.acceleration.y))
{
var half = new Integer(1) / 2;
return
(b.position - a.position - half * a.acceleration * (dt ^ 2))
/
dt;
}
}
throw new Exception();
}
static void Main(string[] args)
{
// A ski jumper leaves the ski track moving in the horizontal
// direction with a speed of 25.0 m/s, as shown in Figure 4.14.
// The landing incline below him falls off with a slope of 35.0°.
// Where does he land on the incline?
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var th = new Symbol("th"); // angle of incline
var objA = new Obj()
{
position = new Point(0, 0),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = 0
};
Func<MathObject, MathObject> numeric = obj =>
obj
.Substitute(vA, 25)
.Substitute(thA, 0.0)
.Substitute(th, (-35).ToRadians())
.Substitute(Trig.Pi, 3.14159)
.Substitute(g, 9.8);
var intersection = objA.ProjectileInclineIntersection(th);
Action nl = () => "".Disp();
"Where does he land on the incline?".Disp(); nl();
"x position (symbolic):".Disp();
intersection.x.Disp(); nl();
"y position (symbolic):".Disp();
intersection.y.Disp(); nl();
"x position (numeric):".Disp();
numeric(intersection.x).Disp(); nl();
"y position (numeric):".Disp();
numeric(intersection.y).Disp(); nl();
var objB = new Obj()
{
position = intersection,
acceleration = _g
};
"Determine how long the jumper is airborne".Disp(); nl();
"symbolic:".Disp();
var timeB = Calc.Time(objA, objB, 1);
timeB.Disp(); nl();
"numeric:".Disp();
numeric(timeB).Disp(); nl();
objB = objA.AtTime(timeB);
"Determine his vertical component of velocity just before he lands".Disp();
nl();
"symbolic:".Disp();
objB.velocity.y.Disp(); nl();
"numeric:".Disp();
numeric(objB.velocity.y).Disp();
Console.ReadLine();
}
static void Main(string[] args)
{
// One strategy in a snowball fight is to throw a first snowball at a
// high angle over level ground. While your opponent is watching the
// first one, you throw a second one at a low angle and timed to arrive
// at your opponent before or at the same time as the first one. Assume
// both snowballs are thrown with a speed of 25.0 m/s. The first one is
// thrown at an angle of 70.0° with respect to the horizontal.
//
// (a) At what angle should the second (low-angle) snowball be thrown
// if it is to land at the same point as the first?
//
// (b) How many seconds later should the second snowball be thrown if it
// is to land at the same time as the first?
var xA = new Symbol("xA"); // position.x at point A
var yA = new Symbol("yA"); // position.y at point A
var th1A = new Symbol("th1A"); // angle of snowball 1 at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
//Func<MathObject, MathObject> numeric = obj =>
// obj
// .Substitute(xA, 0)
// .Substitute(xB, 1.4)
// .Substitute(yA, 0.86)
// .Substitute(g, 9.8)
// .Substitute(Trig.Pi, 3.14159);
var obj1A = new Obj() // snowball 1 at initial point
{
position = new Point(xA, yA),
velocity = Point.FromAngle(th1A, vA),
acceleration = _g,
time = 0
};
var obj1B = new Obj() // snowball 1 at final point
{
position = new Point(null, new DoubleFloat(0.0)),
velocity = new Point(obj1A.velocity.x, null),
acceleration = _g
};
var time1B = Calc.Time(obj1A, obj1B, 1);
obj1B = obj1A.AtTime(time1B);
var obj2A = new Obj() // snowball 2 at initial point
{
position = obj1A.position,
speed = vA,
acceleration = _g
};
var obj2B = new Obj() // snowball 2 at final point
{
position = obj1B.position,
acceleration = _g
};
var th2 = Calc.InitialAngle(obj2A, obj2B, 0, 0);
("At what angle should the second (low-angle) snowball " +
"be thrown if it is to land at the same point as the first?").Disp();
"".Disp();
"symbolic:".Disp();
th2.Disp(); "".Disp();
"numeric:".Disp();
th2
.ToDegrees()
.Substitute(yA, 0)
.Substitute(th1A, (70).ToRadians())
.Substitute(vA, 25)
.Substitute(g, 9.8)
.Substitute(Trig.Pi, Math.PI)
.Disp();
"".Disp();
obj2A.velocity = Point.FromAngle(th2, vA);
var time2B = Calc.Time(obj2A, obj2B, 1);
("How many seconds later should the second snowball be thrown if it " +
"is to land at the same time as the first?").Disp();
"".Disp();
"symbolic:".Disp();
(time1B - time2B).Disp(); "".Disp();
"numeric:".Disp();
(time1B - time2B)
.Substitute(yA, 0.0)
.Substitute(th1A, (70).ToRadians())
.Substitute(vA, 25.0)
.Substitute(Trig.Pi, 3.14159)
.Substitute(g, 9.8)
.Substitute(0, 0.0)
.Disp();
Console.ReadLine();
}
static void Main(string[] args)
{
// A stone is thrown from the top of a building upward at an
// angle of 30.0° to the horizontal and with an initial speed of
// 20.0 m/s, as shown in Figure 4.12. If the height of the building
// is 45.0 m, (a) how long is it before the stone hits the ground?
// (b) What is the speed of the stone just before it strikes the
// ground?
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA = new Obj()
{
position = new Point(0, 0),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = 0
};
var objB = new Obj()
{
velocity = new Point(objA.velocity.x, 0),
acceleration = _g,
};
var timeB = Calc.Time(objA, objB);
objB = objA.AtTime(timeB);
var timeC = timeB * 2;
var objC = objA.AtTime(timeC);
var yD = new Symbol("yD");
var objD = new Obj()
{
position = new Point(null, yD),
velocity = new Point(objA.velocity.x, null),
acceleration = _g
};
var timeAD = Calc.Time(objA, objD, 1);
objD = objA.AtTime(timeAD);
"How long is it before the stone hits the ground?".Disp();
"".Disp();
"Symbolic answer:".Disp();
timeAD.Disp();
"".Disp();
"Numeric answer:".Disp();
timeAD
.Substitute(g, 9.8)
.Substitute(thA, (30).ToRadians())
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 20)
.Substitute(yD, -45)
.Disp();
"".Disp();
"What is the speed of the stone just before it strikes the ground?".Disp();
"".Disp();
"Symbolic answer:".Disp();
objD.velocity.Norm().Disp();
"".Disp();
"Numeric answer:".Disp();
objD.velocity.Norm()
.Substitute(g, 9.8)
.Substitute(thA, (30).ToRadians())
.Substitute(Trig.Pi, 3.14159)
.Substitute(vA, 20)
.Substitute(yD, -45)
.Disp();
Console.ReadKey();
}
static void Main(string[] args)
{
// A fire helicopter carries a 620-kg bucket of water at the
// end of a cable 20.0 m long. As the aircraft flies back
// from a fire at a constant speed of 40.0 m/s, the cable
// makes an angle of 40.0° with respect to the vertical.
// (a) Determine the force of air resistance on the bucket.
// (b) After filling the bucket with sea water, the pilot re-
// turns to the fire at the same speed with the bucket now
// making an angle of 7.00° with the vertical. What is the
// mass of the water in the bucket?
var F1 = new Symbol("F1"); // force of air resistance
var F2 = new Symbol("F2"); // force of cable
var F3 = new Symbol("F3"); // force of gravity
var th1 = new Symbol("th1");
var th2 = new Symbol("th2");
var th3 = new Symbol("th3");
var _F1 = new Point() { angle = th1 };
var _F2 = new Point() { angle = th2 };
var _F3 = new Point() { magnitude = F3, angle = th3 };
var m = new Symbol("m");
var obj = new Obj();
obj.acceleration.x = 0;
obj.acceleration.y = 0;
obj.mass = m;
obj.forces.Add(_F1);
obj.forces.Add(_F2);
obj.forces.Add(_F3);
"Force of air resistance on the bucket:".Disp(); "".Disp();
var FAir =
obj.ForceMagnitude(_F1)
.Substitute(F3, 620 * 9.8)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (90-40).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI)
.Disp();
"".Disp();
_F1.magnitude = FAir;
_F3.magnitude = null;
var FBucketWithWater =
obj.ForceMagnitude(_F3)
.Substitute(th1, (180).ToRadians())
.Substitute(th2, (90 - 7).ToRadians())
.Substitute(th3, (270).ToRadians())
.Substitute(Trig.Pi, Math.PI);
"What is the mass of the water in the bucket?".Disp(); "".Disp();
(FBucketWithWater / 9.8 - 620).Disp();
Console.ReadLine();
}
static void Main(string[] args)
{
// An Alaskan rescue plane drops a package of emergency rations
// to a stranded party of explorers, as shown in Figure
// 4.13. If the plane is traveling horizontally at 40.0 m/s and is
// 100 m above the ground, where does the package strike the
// ground relative to the point at which it was released?
var xA = new Symbol("xA"); // position.x at point A
var yA = new Symbol("yA"); // position.y at point A
var thA = new Symbol("thA"); // angle at point A
var vA = new Symbol("vA"); // velocity at point A
var g = new Symbol("g"); // magnitude of gravity
var _g = new Point(0, -g); // gravity vector
var objA = new Obj() // obj at the initial position
{
position = new Point(xA, yA),
velocity = Point.FromAngle(thA, vA),
acceleration = _g,
time = 0
};
var objB = new Obj() // obj at the final position
{
position = new Point(null, 0),
velocity = new Point(objA.velocity.x, null),
acceleration = _g
};
var timeB = Calc.Time(objA, objB, 1);
objB = objA.AtTime(timeB);
"Where does the package strike the ground relative to the point at which it was released?".Disp(); "".Disp();
"symbolic:".Disp();
objB.position.x.Disp(); "".Disp();
"numeric:".Disp();
objB.position.x
.Substitute(xA, 0)
.Substitute(yA, 100)
.Substitute(vA, 40)
.Substitute(thA, 0.0)
.Substitute(g, 9.8)
.Disp();
("What are the horizontal and vertical components " +
"of the velocity of the package just before it hits the ground?").Disp(); "".Disp();
"symbolic velocity.x:".Disp();
objB.velocity.x.Disp(); "".Disp();
"symbolic velocity.y:".Disp();
objB.velocity.y.Disp(); "".Disp();
"numeric velocity.x:".Disp();
objB.velocity.x
.Substitute(xA, 0)
.Substitute(yA, 100)
.Substitute(vA, 40)
.Substitute(thA, 0.0)
.Substitute(g, 9.8)
.Disp(); "".Disp();
"numeric velocity.y:".Disp();
objB.velocity.y
.Substitute(xA, 0)
.Substitute(yA, 100)
.Substitute(vA, 40)
.Substitute(thA, 0.0)
.Substitute(g, 9.8)
.Disp(); "".Disp();
Console.ReadLine();
}