A superball dropped from the top of Washington Monument (556 ft high) always rebounds ¾..?

Question

A superball dropped from the top of Washington Monument (556 ft high) always rebounds ¾ of the distance fallen. How far (up and down) will the ball have traveled when it hit’s the ground for the 6th time?
Approximate the total distance that the ball of Exercise 65 will have traveled when it comes to rest.

Answer 1

Hi.

It's a simple geometric sequence.

It starts to fall 556 ft, rebounding 3/4 of the way up to 417 ft (3/4 of 556). The sum of n geometric terms is S = a1(1-r^n)/(1-r)

From the first rebound, the ball rises up, and falls the same distance. Thus, if we do not consider the first bounce, the total distance traveled up to the 6th rebound (n=5 because first bounce does not count) is

S = 417(1-(3/4)^n)/(1-3/4)

To get the total height reached, we add up the first height fallen, which is 556, to the sum above.

Now if we consider infinitely many bounces (up until it stops bouncing), we use sum of inifinite terms for geometric sequences which is

S = a1/(1-r)

Since after the first bounce it reaches the same height twice, we have

Total = 556 + 2[417(1-3/4)]

Please do the real computing. ;-)

Answer 2

the Washington Monunent is 555 feet and 3/5 inches tall.

Answer 3

Part 1 -:

556 x 0.75 x 0.75 x 0.75 x 0.75 x 0.75 x 0.75 (i.e 6 bounces)

= 98.956 feet

Part 2-:

1st bounce = 556 (for the initial drop) + 417 (1st rebound)
2nd bounce = 417 (drop) + 312.75 (rebound)
3rd bounce = 312.75 (drop) + 234.5625 (rebound)
4th bounce = 234.5625 (drop) + 175.922 (rebound)
5th bounce = 175.922 (drop) + 131.94 (rebound)
6th bounce = 131.94 (drop) + 98.956 (rebound)
7th bounce = 98.956 (drop) + 74.217 (rebound)
8th bounce = 74.217 (drop) + 55.663 (rebound)
9th bounce = 55.663 (drop) + 41.747 (rebound)
10th bounce = 41.747 (drop) + 31.31 (rebound)
11th bounce = 31.31 (drop) + 23.483 (rebound)
12th bounce = 23.483 (drop) + 17.61 (rebound)
13th bounce = 17.61 (drop) + 13.21 (rebound)
14th " = 13.21 (drop) + 9.91 (rebound)
15th bounce = 9.91 (drop) + 7.43 (rebound)
16th bounce = 7.43 (drop) + 5.57 (rebound)

This is actually an infinite series and this is far enough to go:

Its approximately 3858 feet (so I'd round it up to 4000 feet by the time it is only moving fractionally)

Answer 4

This is just the sum of a geometric sequence (called a series).