A ball is dropped from height of 100 feet towards the ground. It hits the ground and bounces back 80% of the fall. This continues till the ball comes to rest. Find the vertical distance covered by the ball
the height on the nth bounce is 100*0.8^n
So, the geometric sequence for the bounces is
a = 100
r = 0.8
Each bounce is a round trip, except for the 0th trip, so the distance traveled is
2*100/0.2 - 100 = 900 ft
The real-world distance of course will be slightly less than that.
To find the vertical distance covered by the ball, we need to sum up the distances covered during each bounce.
Given:
Initial height (h0) = 100 feet
Bounce efficiency (e) = 80% = 0.8
Let's calculate the distances covered during each bounce and add them up until the ball comes to rest. We can use the formula:
h = h0 + e*h0 + e^2*h0 + e^3*h0 + ...
The distance covered during each bounce is the initial height multiplied by the bounce efficiency raised to the power of the bounce number.
Following the steps:
1. Calculate the distance covered during each bounce:
h1 = e*h0
h2 = e*h1
h3 = e*h2
...
2. Add up the distances until the ball comes to rest:
h_total = h0 + h1 + h2 + h3 + ...
Let's solve it step-by-step:
Step 1: Calculate the distance covered during each bounce:
h1 = 0.8 * 100 = 80 feet
h2 = 0.8 * 80 = 64 feet
h3 = 0.8 * 64 = 51.2 feet
...
Step 2: Add up the distances until the ball comes to rest:
h_total = h0 + h1 + h2 + h3 + ...
h_total = 100 + 80 + 64 + 51.2 + ...
The ball will continue to bounce infinitely, but we can calculate an approximation or a limit for the total distance it covers.
Let's calculate the limit of the infinite sum:
h_total = h0 / (1 - e)
h_total = 100 / (1 - 0.8)
h_total = 100 / 0.2
h_total = 500 feet
Therefore, the vertical distance covered by the ball is approximately 500 feet.
To find the vertical distance covered by the ball, we need to calculate the sum of the distances for each bounce.
Given:
Initial height = 100 feet
Bounce factor = 80% (0.8)
Let's start by calculating the distance covered in the first bounce:
First Bounce:
The ball falls from a height of 100 feet and bounces back 80% of the fall, so it bounces back to a height of 100 * 0.8 = 80 feet.
The total distance covered in the first bounce is the sum of the fall distance and the bounce distance, which is 100 + 80 = 180 feet.
Now, the ball is at a height of 80 feet (from its initial height) after the first bounce.
Second Bounce:
The ball falls from a height of 80 feet and again bounces back 80% of the fall, so it bounces back to a height of 80 * 0.8 = 64 feet.
The total distance covered in the second bounce is the sum of the fall distance and the bounce distance, which is 80 + 64 = 144 feet.
Now, the ball is at a height of 64 feet after the second bounce.
We can continue this process, calculating the distance covered in each subsequent bounce, until the ball comes to rest.
When the ball comes to rest:
The height of each successive bounce decreases, approaching zero. In theory, the ball would keep bouncing infinitely, but for practical purposes, we can assume that the ball comes to rest when the bounce height is negligible (i.e., when it falls to a height close to zero).
To find the total vertical distance covered by the ball, we sum the distances covered in each bounce until the ball comes to rest.
In this case, I have provided you with the calculations for the first two bounces. You can continue the pattern and calculate the distances for each subsequent bounce until the ball comes to rest.