A hard rubber ball is dropped from a moving van with a height of 5 metres. The ball rises to 80% of the previous height after each bounce.

a) determine the total vertical distance the ball has traveled when it strikes the ground the eighth time.

b) estimate the total vertical distance the ball travels before it comes to rest.

It's a geometric series, I know this much. My problem comes from the fact that... the initial 5 metres in an anomaly. After the ball hits the ground the first time, it will go up and down the same distance, then up and down 80% less, etc. It's that first 5 that is an odd distance, it only happens once. How would I write the S_n_ formula?

A few ideas I had were:

Sn=/8(1-.8^n)\
\ 1-.8 / +5

Change a to 2a in the formula, and then subtract a from the end to equalize. This resolves the 1/2 distance on the first bounce.

s8=(2a(1-r^n)/(1-r))-a

To find the total vertical distance the ball has traveled when it strikes the ground the eighth time, we need to consider the distances the ball travels both during the upward and downward bounces.

Let's break it down step by step:

- On the first downward bounce, the ball travels 5 meters.
- After this, the ball rises to 80% of the previous height, which is 5 meters in this case. So, during the first upward bounce, the ball travels 5 * 0.8 = 4 meters.
- During the second downward bounce, the ball will again travel 4 meters.
- During the second upward bounce, the ball will travel 4 * 0.8 = 3.2 meters.

This pattern continues, with each upward bounce covering 80% of the previous height and each downward bounce covering the same distance as the previous downward bounce.

To calculate the total vertical distance the ball has traveled when it strikes the ground the eighth time, we can write the sum of these distances as a geometric series:

S_8 = 5 + 4 + 4 * 0.8 + 4 * 0.8^2 + ... + 4 * 0.8^7

The formula for the sum of a geometric series is:

S_n = a * (1 - r^n) / (1 - r)

In this case, a = 5 (the initial distance) and r = 0.8 (the common ratio).

So, we can calculate S_8 as follows:

S_8 = 5 * (1 - 0.8^8) / (1 - 0.8)

Simplifying this expression will give us the total vertical distance the ball has traveled when it strikes the ground the eighth time.

For the second question, estimating the total vertical distance the ball travels before it comes to rest, we need to consider that the height of each bounce decreases by 80% of the previous height.

In this case, since the ball will eventually come to rest, the total vertical distance traveled can be approximated as the sum of an infinite geometric series. We can use the formula:

S = a / (1 - r)

Again, a = 5 (the initial distance) and r = 0.8 (the common ratio).

S = 5 / (1 - 0.8)

Simplifying this expression will give us an estimation of the total vertical distance the ball travels before it comes to rest.

It's important to note that these calculations assume ideal conditions and may not perfectly reflect the real-world behavior of the ball.