Question

A tennis ball bounces so that its initial speed straight upwards is b feet per
second. Its height s in feet at time t seconds is given by s = bt − 16t
2
a) Find the velocity v = ds/dt at time t.
b) Find the time at which the height of the ball is at its maximum height.
c) Find the maximum height.
d) Make a graph of v and directly below it a graph of s as a function of time.
Be sure to mark the maximum of s and the beginning and end of the bounce.

Answer 1

e) Suppose that when the ball bounces a second time it rises to half the
height of the first bounce. Make a graph of s and of v of both bounces, labelling
the important points. (You will have to decide how long the second bounce lasts
and the initial velocity at the start of the bounce.)
f) If the ball continues to bounce, how long does it take before it stops?

Answer 2

a) \[\lim_{\Delta t \rightarrow 0}(b(t+\Delta t)-16(t+\Delta t^2)-(bt-16t^2)/\Delta t\]=b-32t b) At maximum height, velocity is equal to zero since its changing directions t=b/32 seconds c) s=b^2/32-b^2/64=b^2/64

Answer 3

If the initial velocity of the first bounce is b1, maximum height is b1^2/64 and time for maximum for the end of the first bounce is b1/16. This implies that the maximum height for the second bounch is b2^2/64 which is 1/2 the maximum height of the first.
b1^2/64*1/2=b2^2/64--> b1/root(2)=b2---> b1=root(2)b2 and hence the total time of the first and second would be b1/16+b1/16 root(2).