To find how far the ball will travel before coming to rest, we can calculate the sum of the distances it travels while falling and rebounding.
Given:
Initial height (h) = 48 m
Rebound distance ratio = 2/3
Step 1: Calculate the distance the ball travels on the first fall.
Distance of the first fall = Initial height (h) = 48 m
Step 2: Calculate the distance of the first rebound.
Distance of the first rebound = Rebound distance ratio * distance of the first fall
Distance of the first rebound = (2/3) * 48 m = 32 m
Step 3: Calculate the distance the ball travels on the second fall, using the starting point of the first rebound.
Distance of the second fall = Distance of the first rebound
Distance of the second fall = 32 m
Step 4: Calculate the distance of the second rebound.
Distance of the second rebound = Rebound distance ratio * distance of the second fall
Distance of the second rebound = (2/3) * 32 m = 21.33 m (rounding to two decimal places)
Step 5: Calculate the distance the ball travels on the third fall, using the starting point of the second rebound.
Distance of the third fall = Distance of the second rebound
Distance of the third fall = 21.33 m
Step 6: Calculate the distance of the third rebound.
Distance of the third rebound = Rebound distance ratio * distance of the third fall
Distance of the third rebound = (2/3) * 21.33 m = 14.22 m (rounding to two decimal places)
Continue this process until the distance of the rebound becomes negligible (approaching zero).
Now, we need to find the sum of all the distances traveled by the ball.
Sum of distances = Distance of first fall + Distance of first rebound + Distance of second fall + Distance of second rebound + Distance of third fall + Distance of third rebound + ...
Sum of distances = 48 m + 32 m + 32 m + 21.33 m + 21.33 m + 14.22 m + 14.22 m + ...
To find the sum, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r)
Where:
a = Distance of first fall = 48 m
r = Rebound distance ratio = 2/3
Sum = 48 m / (1 - 2/3)
Simplifying,
Sum = 48 m / (1/3)
Sum = 48 m * (3/1)
Sum = 48 m * 3
Sum = 144 m
Therefore, the ball will travel a total distance of 144 meters before coming to rest.