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 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)
{
// 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?
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 xB = new Symbol("xB");
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 objA = new Obj()
{
position = new Point(xA, yA),
velocity = new Point(null, 0),
acceleration = _g,
time = 0
};
var objB = new Obj()
{
position = new Point(xB, 0),
velocity = new Point(null, null),
acceleration = _g
};
var timeB = Calc.Time(objA, objB, 1);
objB.time = timeB;
var _vA = Calc.InitialVelocity(objA, objB);
objA.velocity = _vA;
"With what velocity did the mug leave the counter?".Disp(); "".Disp();
"symbolic:".Disp();
"x:".Disp(); _vA.x.Disp();
"y:".Disp(); _vA.y.Disp();
"".Disp();
"numeric:".Disp();
"x:".Disp(); numeric(_vA.x).Disp();
"y:".Disp(); numeric(_vA.y).Disp();
"".Disp();
"What was the direction of the mug’s velocity just before it hit the floor?".Disp(); "".Disp();
objB = objA.AtTime(timeB);
"symbolic:".Disp();
objB.velocity.ToAngle().Disp(); "".Disp();
"numeric:".Disp();
numeric(objB.velocity.ToAngle().ToDegrees()).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)
{
// 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();
}
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();
}
