Maximum height of a ball: The position of a ball thrown from a cliff is given by

We can determine the maximum height of the ball by differentiating, setting the derivative to 0, and solving for the time, tmax. Then the maximum height is found by using tmax in y(t):
. First find tmax:

Setting this to 0, we get 0 = 64 - 32 tmax, or tmax = 2.

Now, we can find the maximum height:

ft

Note the physical meaning of . It means that the velocity is momentarily 0 at the highest point on the path. Note also that

ft/sec/sec. This negative value of the second derivative is consistent with the occurrence of a maximum.

Test Problem: Solve the same problem with

tmax ymax
a) 0.5 96.25
b) 4.0 60
c) 1.94 15.2
d) 2.0 130