The total vertical distance that the ball travels is 84m.
This scenario involves an infinite geometric series! We can separate the ball's movement into two series: one for its downward falls and another for its upward bounces.
Downward Falls:
Starting from 12 m, each fall is 3/4 of the previous height. This forms a geometric series with:
First term (a) = 12 m
Common ratio (r) = 3/4
Upward Bounces:
Starting from 9 m (3/4 of 12 m), each bounce is also 3/4 of the previous height. This forms another geometric series with:
First term (a) = 9 m
Common ratio (r) = 3/4
Now, we use the formula for the sum of an infinite geometric series: S = a / (1 - r).
Downward Distance:
S_down = 12 / (1 - 3/4) = 48 m
Upward Distance:
S_up = 9 / (1 - 3/4) = 36 m
Total Distance:
To find the total vertical distance, we add the distance traveled in both directions:
Total Distance = S_down + S_up = 48 m + 36 m = 84 m
Therefore, the ball travels a total vertical distance of 84 meters before coming to rest.
Question:
A ball is dropped from a height of 12m. After each bounce, it rises to 3/4 of the height of the previous bounce. What is the total vertical distance that the ball travels?