Question

Problem Set 1
Question 1B-2(e):

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)

My question is part (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, labeling
the important points. (You will have to decide how long the second bounce lasts and the initial velocity at the start of the bounce.)

Answer 1

Okay so, I understand how to get the velocity of the 2nd bounce but I am having trouble figuring out how long the second bounce lasts.

The solution says the second bounce is at (b/16) + (b/16)

If the 2nd bounce height is half of the first bounce, wouldn't it take HALF the time of the first bounce which is just (b/16) divided by 2?

Answer 2

The two velocities at the start of bounce 1 and bounce 2 are not the same...

Answer 3

oh wow, how did I miss that lol. I completely missed that it was \[b _{1}/16 + b _{2}/16\]
but I'm still confused on why it is over 16? If \[b _{1}/16 \] is the time the ball was in the air for the First bounce then how do you know the second bounce is \[b _{2}/16\]
Thanks for any help!

Answer 4

If you go back to part c) on the same problem, you'll notice that the maximum height of a bounce b_1 is \[b_1^2/64\]on the first bounce. Then part e) says that the second bounce with initial velocity \[b_2\]is half the height of the first bounce. Therefore we can come up with this equality \[(1/2)(b_1^2/64) = b_2^2/64\] Solving for b_1 we get \[b_1 = b_2\sqrt2\] But this doesnt really help with getting that the total time is \[b_1/16+b_2/16\] does it? But notice that the time t will always be over 16, because its the same equation, just a different variable. If you wanted the put the time of the second bounce in terms of the velocity of the first bounce, then you would use the relation between the velocities of the two bounces to get total time = \[b_1/16+b_2/16 = b_1/16 + b_1/16\sqrt2\] Hope I helped, and please don't hesitate to ask for clarification!

Answer 5

Hey thanks a lot for the help thus far.
So does \[b _{1}/16 + b _{2}/16 \] represent the time the ball was in the air for the SECOND bounce or is it the total of time the ball was in the air in the first AND second bounce?

Answer 6

Total time for first + second bounce.

Answer 7

okay thanks. I'm going to try and mull things over and hopefully it will be clear to me. Thanks a lot, Nitro!

Answer 8

you're welcome!

Answer 9

Just a quick question but could \[b _{2}/16\] be the time of from the beginning of the second bounce to the end?

Answer 10

Thats exactly what it is