s
s = s 0
s = 0
Math 163, Spring 2001 Due Date: Name(s):
Project 3.4: Ballistics
Objective
To illustrate an important application of differentiation to ballistics.
Narrative
If you have not already done so, do Project 3.8 Differentiation. In that project we illustrate how derivatives
can be computed in Maple.
If a projectile is fired vertically upward with an initial velocity
of v0 m/sec from an initial position s0 meters above
the ground (see the figure to the right), 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, and the velocity of the projectile,
after t seconds, is
v = v(t) = Dt(s(t)) = -gt + v0
meters per second. (If s is measured in feet ft and v is
measured in ft/sec, then g = 32 ft/sec2
.)
Task
a) Type the command lines in the left-hand column below into Maple in the order in which they are listed.
> # Project 3.4: Ballistics
> restart; Clear Maple's memory.
> g := 9.8; t0 := 0; s0 := 100; v0 := 128; Let g = 9.8, t0 = 0, s0 = 100, and v0 = 128.
(In this project we'll be using metric units.)
> s := t -> -0.5*g*t^2+v0*t+s0; Let the distance s(t) = -1
2 gt2
+ v0t + s0.
> plot(s(t),t=t0..20); Graph s(t) for t  [t0, 20]. Observe that after
20 sec, the projectile is still in the air.
Let's find when the projectile hits the ground.
> solve(s(t)=0,t); Find when s(t) = 0. You should get two values:
one negative and one positive. The positive value
is the time at which the projectile hits the ground.
> t1 := %[2]; Let t1 be the the positive value. (We're assuming
here that the second value is positive; if it's the
first value that's positive, type t1 := %[1];
instead.)
> plot(s(t),t=t0..t1); Graph s(t) for t  [t0, t1].
> v := D(s); Let the velocity v(t) = Dt(s(t)).
> v(t1); Find the velocity of the projectile when it hits
the ground.
> t smax := solve(v(t)=0,t); Find the time tsmax at which the velocity of the
projectile is 0; tsmax is the time it takes the
projectile to reach its maximum altitude s(tsmax).
> v(t smax); This just checks Maple's work: the result should be
zero (or close to zero).
> s(t smax); Find the maximum altitude s(tsmax) of the
projectile.
> plot(v(t),t=t0..t1); Plot v as a function of t for t  [t0, t1].
At this point, make a hard-copy of your typed input and Maple's responses. Then ...
b) Label by hand the coordinate axes in the second graphic you produced. (One should be a t-axis, and the
other an s-axis.) Plot and label the points (t, s(t)) for t = t0, t = t1, and t = tsmax in this graphic.
c) Label by hand the coordinate axes in the third graphic you produced. (One should be a t-axis, and the
other a v-axis.) Plot and label the points (t, v(t)) for t = t0, t = t1, and t = tsmax in this graphic.
Your lab report will be a hard copy of your typed input and Maple's responses (both text and hand-labeled
graphics).