A superball is dropped from a height of 2 m and bounces 90% of its original height on each bounce.

(a) When it hits the ground for the eighth time, how far has it traveled?
(b) How high off the floor is the ball on the eighth bounce?

Angie's right. The ball needs to fall down after the 7th bounce, i.e. we stop counting after the 8th down arrow, to the sum 19.84, we need to add 0.96.

This is actually a summation of a geometric series of common ratio r=0.9, and initial value of 2.0.
The total (up and down) distance is twice the sum less 2 because the ball did not bounce for the first time.
The sum S(n) of the geometric series is
S(n)=a(1-r^n)/(1-r)
So for n=8,
S(8)=2(1-0.9^8)/(1-0.9)
=11.39066
So the total distance travelled is twice the sum less 2 (initial up motion)
=2*S(8)-2
=20.78

For part (b), the height h of the ball after the nth bounce is ar^n, where a=initial height, r=rebound factor, and n=number of bounces.
For a=2, r=0.9, n=8,
h=0.9^8=0.86

1. 2↓,

2. 1.8↑, 1.8↓,
3. 1.62↑,1.62↓.
4. 1,46↑, 1.46↓,
5. 1.31↑, 1.31↓,
6. 1.18↑ , 1.18↓,
7. 1.07↑, 1.07↓,
8. 0.96↑,
(a) Σ =19.84 m.
(b) 0.96 m

the question is asking for the eighth time it hits the ground so wouldn't that include the other part of the .96, because what you have here is technically hitting the ground 7 times and the ball is in mid-air.... and for b) how did you set the problem up?

Thank you!

Thank you for the clarification on A and for B the way the question is worded I think I'm going to go with the .96...

THANKS GUYS!

You're welcome!

Note:
If it is a trick question, the height of the ball on the eighth bounce is zero!

To calculate the distance traveled by the superball when it hits the ground for the eighth time, we need to determine the total distance covered during each bounce and then sum it up.

(a) The first step is to calculate the distance traveled during each bounce. Since the superball bounces back to 90% of its original height, the distance it covers after each bounce can be calculated as follows:

Distance covered = 2 * original height * 0.9

In this case, the original height is 2m, so the distance covered during each bounce is:

Distance covered = 2 * 2 * 0.9 = 3.6m

Now, to calculate the total distance traveled for all eight bounces, we multiply the distance covered during each bounce by the number of bounces:

Total distance traveled = distance covered * number of bounces
= 3.6 * 8
= 28.8m

Therefore, when the superball hits the ground for the eighth time, it has traveled a total distance of 28.8m.

(b) To find out how high off the floor the ball is on the eighth bounce, we need to calculate the height after each bounce.

Height after each bounce = original height * (bounce factor)^number of bounces

In this case, the original height is 2m, and the bounce factor is 0.9. So, the height after the eighth bounce can be calculated as:

Height after eighth bounce = 2 * (0.9)^8
≈ 2 * 0.43
≈ 0.86m

Therefore, the ball is approximately 0.86m high off the floor on the eighth bounce.