A ball rebounds the ground to a height of 72 cm, and each time it rebounds to two thirds of the height. Estimate the total distance the ball travels before it comes to rest
counting the first bounce at up and down ...
we have
144 + 144(2/3) + 144(2/3)^2 + ...
which is an infinite geometric series where
a = 144 , r = 2/3
sum(all terms) = a/(1-r)
= 144/(1-2/3)
= 144/(1/3)
= 432 cm
To estimate the total distance the ball travels before it comes to rest, we can use a geometric series formula.
First, let's find the total number of bounces the ball makes before it comes to rest. Since the ball rebounds to two-thirds of its previous height, we can calculate the number of bounces using the formula:
Height = Initial Height * (Rebound Height Ratio)^(Number of Bounces)
Where:
- Initial Height is the height from which the ball is dropped (in this case, 72 cm).
- Rebound Height Ratio is two-thirds (or 2/3).
Now, let's solve for the number of bounces:
72 = 72 * (2/3)^Number of Bounces
By dividing both sides by 72, we get:
1 = (2/3)^Number of Bounces
To solve for the number of bounces, we can take the logarithm of both sides:
log(1) = log((2/3)^Number of Bounces)
Using the logarithmic property log(a^b) = b * log(a), the equation becomes:
0 = Number of Bounces * log(2/3)
Now, let's solve for the number of bounces:
Number of Bounces * log(2/3) = 0
Since log(2/3) is a negative value, the only solution for the equation above is:
Number of Bounces = 0
This means that the ball comes to rest after the initial drop. Therefore, the total distance the ball travels before coming to rest is equal to the distance it traveled during the initial drop, which is 72 cm.