a ball is dropped from a height of 100 ft. Each time it hits the floor, it rebounds to 2/3 its previous height. Find the total distance traveled.
Calculate the limit of the series
100[1 + 2*(2/3) + 2*(2/3)^2 + 2*(2/3)^3+...]
= 100 + 2[1 + 2/3 + (2/3)^2+ ...]
Make use of the relation:
1 + x + x^2 + .. = 1/(1-x)
(for x<1, which is the case here)
= 100 [1 + (4/3)[1 + 2/3 + (2/3)^2+ ...]
= 100 {1 + (4/3)[1/(1 - 2/3)]} = ?
To find the total distance traveled by the ball, we need to consider both the distance covered while falling and while rebounding.
Let's break down the problem step by step:
Step 1: Calculate the distance traveled while falling.
When the ball is dropped from a height of 100 ft, it falls until it hits the ground. We can calculate this distance using the formula for distance fallen due to gravity:
Distance fallen = 0.5 * acceleration due to gravity * (time)^2
In this case, the acceleration due to gravity is constant at 32 ft/s^2, and the ball is dropped from rest, so the initial velocity is 0 ft/s. Let's assume the ball takes time "t" seconds to hit the ground.
Since distance = initial velocity * time + 0.5 * acceleration due to gravity * (time)^2, and initial velocity = 0, we can simplify the formula to:
Distance fallen = 0.5 * acceleration due to gravity * (time)^2
Substituting the given values:
Distance fallen = 0.5 * 32 * (t)^2
Step 2: Calculate the distance traveled while rebounding.
Each time the ball rebounds, it reaches 2/3 of its previous height. So, the distance traveled during rebounding is 2/3 of the distance fallen in the previous fall.
Distance rebounding = (2/3) * (previous distance fallen)
Since the ball was initially dropped from a height of 100 ft, the first distance fallen is 100 ft.
Step 3: Calculate the total distance traveled.
To find the total distance traveled, we need to consider the distance traveled while falling and rebounding.
Since the ball rebounds multiple times, we have a geometric series. The total distance can be calculated using the formula for the sum of an infinite geometric series:
Total distance traveled = distance fallen + distance rebounding + distance rebounding^2 + distance rebounding^3 + ...
Since the distance rebounding forms a geometric series with a common ratio of 2/3, we can apply the formula for the sum of an infinite geometric series:
Total distance traveled = distance fallen / (1 - ratio)
Total distance traveled = distance fallen / (1 - 2/3)
Total distance traveled = distance fallen / (1/3)
Simplifying, we have:
Total distance traveled = 3 * distance fallen
Substituting the value of distance fallen from Step 1:
Total distance traveled = 3 * (0.5 * 32 * (t)^2)
Total distance traveled = 48 * (t)^2
Therefore, the total distance traveled by the ball is 48 * (t)^2.