College Algebra 9th txtbk

A-26 APPENDIX B SPECIAL TOPICS
Z Projectile Motion
Newton’s laws and advanced mathematics can be used to determine the path of a projectile.
If v 0 is the vertical speed of the projectile, h 0 is the horizontal speed, and a 0 is the initial
altitude of the projectile (Fig. 4), then, neglecting air resistance, the path of the projectile
is given by
x h 0 t
y a 0 v 0 t 4.9t 2 0 t b
(5)
y
v 0
v 0
a 0
h 0
x
Z Figure 4 Projectile motion.
The parameter t represents time in seconds, and x and y are distances measured in meters.
Solving the first equation in equations (5) for t in terms of x, substituting into the second
equation, and simplifying produces the following equation:
h 2 x 2
0
y a 0 v 0
x 4.9
h 0
You should verify this by supplying the omitted details.
We recognize equation (6) as a parabola. This equation in x and y describes the path
the projectile follows but tells us little else about its flight. On the other hand, the parametric
equations (5) not only determine the path of the projectile but also tell us where
it is at any time t. Furthermore, using concepts from physics and calculus, the parametric
equations can be used to determine the velocity and acceleration of the projectile at
any time t. This illustrates another advantage of using parametric representations of plane
curves.
(6)
EXAMPLE 2 Projectile Motion
An automobile drives off a 50-meter cliff traveling at 25 meters per second (Fig. 5). When
(to the nearest tenth of a second) will the automobile strike the ground? How far (to the
nearest meter) from the base of the cliff is the point of impact?
50 m
Z Figure 5