To find the horizontal distance that a bouncing ball has traveled, you can use a geometric series formula. Given that the ball is dropped from a height of 4 meters and each bounce is 0.68 times the height of the previous bounce, we can calculate the total horizontal distance covered by summing up the distances traveled during each bounce.
Let's denote the initial height as H and the bounce height as B (where B = 0.68 * H). The distance traveled during each bounce is twice the horizontal distance covered during the descent from the bounce height.
To calculate the total horizontal distance, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r),
where:
- a is the first term of the series (the initial horizontal distance traveled during the descent),
- r is the common ratio between successive terms (the horizontal distance traveled during each subsequent descent as a fraction of the previous descent).
In this case, a = 2H (since the horizontal distance traveled during the descent is twice the height of the bounce), and r = 0.68.
Let's plug in the values and calculate:
H = 4 (initial height) B = 0.68 * H = 0.68 * 4 = 2.72 (bounce height) a = 2 * H = 2 * 4 = 8 (initial horizontal distance) r = B / H = 2.72 / 4 = 0.68 (common ratio)
Sum = a / (1 - r) = 8 / (1 - 0.68) = 8 / 0.32 = 25
Therefore, the total horizontal distance that the bouncing ball has traveled is 25 meters.