When dropped on a hard surface, a rubber ball takes a series of bounces, each one about 4/5 as high as the preceding one. If the rubber ball is dropped from a height of 20 ft, what is the approximate distance the ball travels before coming to rest? Please explain how you got the answer. Thanks!
assume an infinite number of hops. In practice, of course, the ball stops earlier than that, but you have a geometric progression where
a = 20
r = 4/5
So, the initial drop is 20, and each bounce is a round trip 4/5 as high as the one before, so the total distance after n hops is
20 + 2(20 * 4/5) + 2(20*4/5)^2 + ...
So, the real GP is
a = 40
r = 4/5
and 20 must be subtracted because the first hop did not have to go up.
The total distance is thus
40/(1 - 4/5) - 20 = 180 ft
Thank you so much!
To find the approximate distance the ball travels before coming to rest, we need to sum up the distances covered during each bounce.
Given that each bounce is about 4/5 as high as the preceding one, we can say that the height of each bounce is 4/5 of the previous bounce.
Let's calculate the height of each bounce using the formula:
Height of bounce = (4/5) * previous bounce height
Given that the initial drop height is 20 ft, we can write the heights of each bounce as follows:
1st bounce: 20 ft
2nd bounce: (4/5) * 20 ft
3rd bounce: (4/5) * (4/5) * 20 ft
4th bounce: (4/5) * (4/5) * (4/5) * 20 ft
The general formula to calculate the height of the nth bounce is:
Height of nth bounce = (4/5)^(n-1) * 20 ft
Now, we need to add up the distances traveled during each bounce. The distance traveled is twice the height of each bounce since the ball covers the same distance going up and coming back down.
So, the distance traveled during each bounce is:
Distance of bounce = 2 * height of bounce
Using the height formula mentioned above, the distance of each bounce can be written as:
Distance of nth bounce = 2 * (4/5)^(n-1) * 20 ft
Now, let's calculate the total distance covered by summing up all the distances of each bounce until it comes to rest. We need to find the last bounce where the height becomes negligible.
Let's calculate the height of the bounce until it becomes negligible when it is less than 1 ft.
(4/5)^(n-1) * 20 ft < 1 ft
Dividing both sides by 20 ft:
(4/5)^(n-1) < 1/20
Taking the base 4/5 logarithm of both sides:
(n-1) < log(1/20) / log(4/5)
Simplifying:
(n-1) < log(1/20) / log(4/5)
(n-1) < -2.301 / -0.322
(n-1) < 7.13
Since n has to be a positive integer, the last bounce will be when n = 8.
Now, let's calculate the total distance traveled:
Total distance = Distance of 1st bounce + Distance of 2nd bounce + ... + Distance of 8th bounce
Total distance = [2 * (4/5)^(0) * 20] + [2 * (4/5)^(1) * 20] + ... + [2 * (4/5)^(7) * 20]
Total distance = 40 + 32 + 25.6 + 20.48 + 16.38 + 13.10 + 10.48 + 8.38
Total distance ≈ 166.42 ft
Therefore, the approximate distance the ball travels before coming to rest is approximately 166.42 ft.
To find out the approximate distance the rubber ball travels before coming to rest, we can consider the sum of all the distances covered during each bounce.
Given that each bounce is approximately 4/5 as high as the preceding one, we can express this mathematically as a geometric series with a common ratio of 4/5.
Let's break down the distances covered during each bounce:
1st bounce: The ball is dropped from a height of 20 ft.
2nd bounce: The ball reaches a height of (4/5) * 20 ft.
3rd bounce: The ball reaches a height of (4/5)^2 * 20 ft.
...
nth bounce: The ball reaches a height of (4/5)^(n-1) * 20 ft.
We can see that the height of each bounce decreases exponentially with each subsequent bounce.
Since the ball eventually comes to rest, we want to find the sum of the distances covered by adding up the distances of each bounce until the height becomes negligible (approaching 0).
Mathematically, the sum of an infinite geometric series can be calculated using the formula:
Sum = a / (1 - r),
where "a" is the first term and "r" is the common ratio.
In this case, the first term "a" is 20 ft, and the common ratio "r" is 4/5.
Using the formula, we can calculate the sum:
Sum = 20 / (1 - 4/5)
Sum = 20 / (1/5)
Sum = 20 * 5
Sum = 100 ft
Therefore, the rubber ball travels approximately 100 ft before coming to rest when dropped from a height of 20 ft.