A tennis ball is dropped from a height of 7 ft. If the ball rebounds 2/3 of its height on each bounce, how far will the ball travel before coming to rest? (Find the total distance that the ball will travel before it stops bouncing.)
i need help i don't know which formula to use to solve this problem.
in theory, the ball will never come to rest, since its bounce height is never zero.
But, you can see the height of the nth bounce is 7(2/3)^n
So, since the distance traveled is both up and down, the sequence is
7 + 2*7(2/3) + 2*7(2/3)^2 + ...
so, if r=2/3,
= 7 + 14(r+r^2+...)
After n bounces the sum is
7 + 14((1-r^n)/(1-r) - 1)
and as n->infinity, that leaves
7 + 14(3/2 - 1) = 7+7 = 14
I think ...
To solve this problem, you can use the formula for the sum of a geometric series. In this case, the series represents the distances travelled by the tennis ball on each bounce.
The formula for the sum of a geometric series is as follows:
S = a * (1 - r^n) / (1 - r)
Where:
S = sum of the series
a = first term of the series
r = common ratio between the terms
n = number of terms in the series
In this case, the first term (a) is the initial height of the ball (7 ft), the common ratio (r) is 2/3 (as the ball rebounds 2/3 of its height on each bounce), and the number of terms (n) is the number of bounces before the ball comes to rest.
Since the ball comes to rest when it reaches a height less than or equal to 1/1000 (or 0.001) ft, we need to find the number of bounces (n) that satisfies this condition.
Let's calculate the number of bounces (n) using the formula:
n = log(1/1000) / log(2/3)
Using a calculator, we find that n is approximately 10.08 (rounding up to the nearest whole number).
Now, let's use the formula for the sum of the series to find the total distance traveled by the ball:
S = 7 * (1 - (2/3)^10) / (1 - 2/3)
Using a calculator, we find that S is approximately 33.005 ft.
Therefore, the ball will travel approximately 33.005 ft before coming to rest.
To solve this problem, you don't necessarily need to use a specific formula. Instead, you can use a logical approach to determine the total distance traveled by the ball before it comes to rest.
Here's how you can break it down step by step:
1. The ball is dropped from a height of 7 ft, so the initial drop can be considered as the first bounce.
2. The ball rebounds 2/3 of its height on each bounce, which means it will reach a height of (2/3) * 7 ft = 14/3 ft on the first bounce.
3. The ball will continue bouncing until its rebound height becomes less than the height of a single bounce. In this case, it will stop bouncing when the rebound height is less than 7 ft.
4. To find the total distance traveled by the ball before coming to rest, you need to sum up the height of each bounce.
Now, let's calculate it:
- The first bounce covers a total distance of 7 ft (drop height).
- The second bounce covers a total distance of twice the rebound height, which is 2 * (14/3) ft.
- The third bounce covers a total distance of twice the rebound height, which is 2 * (2/3) * (14/3) ft.
- Similarly, the distances covered by subsequent bounces can be calculated using the same logic until the rebound height becomes less than 7 ft.
You can continue this process and sum up the distances covered in each bounce until the height becomes less than 7 ft. This will give you the total distance the ball will travel before coming to rest.