x
y
y 0
MATH 164, Spring 2001 Due Date: Name(s):
Project 11.1c: Ballistics
Objective
To illustrate the application of parametric curves to ballistics.
Narrative
If a projectile is fired vertically upward into the air with an initial velocity of v0 m/sec from a point s0
meters above the ground, then (neglecting air resistance) after t sec the projectile is
s = s(t) = -
1
2
gt2
+ v0t + s0
meters above the ground, where g = 9.8 m/sec2
is acceleration
due to gravity. If the projectile is fired at an angle
of elevation of  with respect to the horizontal at an initial
velocity of v0 m/sec from a point y0 meters above the
ground (see the figure to the right), then (neglecting air
resistance) after t sec the projectile is located at the point
whose coordinates are given parametrically by
x(t) = (v0 cos )t, y(t) = -
1
2
gt2
+ (v0 sin )t + y0.
Task
a) Type the command lines below into Maple in the order in which they are listed. They describe the motion
of a projectile that is fired at time t0 = 0 at an angle of elevation of  = /4 with respect to the horizontal
at an initial velocity of v0 = 128 m/sec from a point y0 = 100 meters above the ground.
> # Project 11.1c: Ballistics
> restart;
> g := 9.8; t0 := 0; y0 := 100; v0 := 128; theta := Pi/4;
> x := t -> v0*cos(theta)*t;
> y := t -> -0.5*g*t^2+v0*sin(theta)*t+y0;
> vx := D(x); vy := D(y);
> t1 := 1.0; evalf(y(t1)); plot([x(t),y(t),t=t0..t1]);
(The quantities vx and vy are known as the x- and y-components of the velocity, respectively. We discuss
these quantities further in MATH 261.)
b) By using trial-and-error, change the value of t1 in the last line you typed until you obtain the value of t1
greater than t0 for which y(t1) is within 2 decimal places of 0. (In doing this you are estimating the time
t1 it takes the projectile to hit the ground.)
c) Continue by typing the command lines in the left-hand column below into Maple in the order in which
they are listed.
> evalf(x(t1)); This is the range of the projectile.
> evalf(sqrt(vx(t1)^2+vy(t1)^2)); This is the impact velocity. (We discuss impact velocity
further in MATH 261.)
> t max := solve(vy(t)=0,t); This is the time at which the vertical component of the
velocity is 0; this is the time at which the projectile
achieves its maximum altitude.
> evalf(y(t max)); This is the maximum altitude.
The range of a projectile is the horizontal (or x-) distance the projectile travels before it strikes the ground.
To compute the range of a projectile that is fired at an angle of elevation of  with respect to the horizontal
at an initial velocity of v0 m/sec from a point y0 meters above the ground in closed form, observe that at
the time t1 the projectile strikes the ground,
y(t1) = -
1
2
gt2
1 + (v0 sin )t1 + y0 = 0.
Thus t1 = (v0 sin  + v2
0 sin2
 + 2gy0)/g, and the range
x(t1) = (v0 cos )t1 = v0 cos 
v0 sin  + v2
0 sin2
 + 2gy0
g
.
If, in particular, y0 = 0, then the range
x(t1) =
v2
0 sin 2
g
.
d) What is the maximum range of a projectile that is fired with initial velocity v0 from ground level y0 = 0?
At what angle  is this maximum range achieved?
e) If a projectile with an initial velocity of 128 m/sec is fired from ground level, can it hit a target 1600
meters away? Justify your answer. If the projectle can hit the target, at what angle of elevation  must it
be fired so that it will hit the target?
f) Repeat part (e) for a target that is 1800 meters away.
Your lab report will be a hard copy of your typed input and Maple's responses, as well as your answers to
parts (d), (e), and (f).
Comments
There are numerous other questions about ballistics that you are now able to handle and that might be
of interest to you. For further information, consult your instructor!