A ball is dropped from a height of 18 meters and each rebound is 1/3 of the previous distance. What is the total distance the ball has travelled before coming to the rest
To find the total distance the ball has traveled before coming to rest, we need to sum up the distances traveled during each rebound.
Let's start by finding the distances traveled during each rebound:
Rebound 1: 18 meters (the initial drop)
Rebound 2: 18 * 1/3 = 6 meters
Rebound 3: 6 * 1/3 = 2 meters
Rebound 4: 2 * 1/3 = 0.67 meters
Rebound 5: 0.67 * 1/3 = 0.22 meters
(We stop here because the distance becomes negligible/insignificant)
Now let's add up these distances:
Total distance = 18 + 6 + 2 + 0.67 + 0.22
Total distance = 27.89 meters
Therefore, the ball has traveled approximately 27.89 meters before coming to rest.
To calculate the total distance the ball has traveled before coming to rest, we need to sum up the distance traveled during each rebound.
First, let's calculate the distance traveled during the first rebound. The ball is initially dropped from a height of 18 meters, so the first rebound covers a distance of 18 meters.
For each subsequent rebound, the distance covered is 1/3 of the previous rebound's distance. So, the distances covered during the rebounds are as follows:
Rebound 1: 18 meters (initial drop)
Rebound 2: 1/3 * 18 = 6 meters
Rebound 3: 1/3 * 6 = 2 meters
Rebound 4: 1/3 * 2 = 2/3 meters
Rebound 5: 1/3 * 2/3 = 2/9 meters
...
Rebound n: 1/3^n * 18 meters
To find the total distance traveled, we need to sum up distances traveled during each rebound. We can use a geometric series formula to calculate the sum.
The formula for the sum of an infinite geometric series is:
S = a / (1 - r)
Where a is the first term and r is the common ratio.
In this case, a = 18 meters and r = 1/3.
Plugging in the values into the formula:
S = 18 / (1 - 1/3) = 18 / (2/3) = 18 * (3/2) = 27 meters
Therefore, the total distance traveled by the ball before coming to rest is 27 meters.
To find the total distance the ball has traveled before coming to rest, we need to consider both the upward and downward motion of the ball during each rebound.
Given that the ball is dropped from a height of 18 meters, let's analyze its motion during each rebound. The distance traveled during each rebound is one-third of the previous distance.
1st rebound:
The ball rebounds one-third of the distance it fell and reaches a height of 18 * (1/3) = 6 meters.
The total distance traveled so far is 18 (initial drop) + 6 (rebounce) = 24 meters.
2nd rebound:
The ball rebounds one-third of the distance it fell from the previous rebound, which is one-third of 6 meters. So, it reaches a height of 6 * (1/3) = 2 meters.
The total distance traveled so far is 24 + 6 + 2 = 32 meters.
3rd rebound:
The ball rebounds one-third of the distance it fell from the previous rebound, which is one-third of 2 meters. So, it reaches a height of 2 * (1/3) = 2/3 meters.
The total distance traveled so far is 32 + 2 + 2/3 ≈ 34.67 meters.
Each subsequent rebound will get smaller and smaller until it reaches a negligible distance, approaching zero. Therefore, we can approximate the total distance the ball traveled before coming to rest by summing up the distances traveled during the first three rebounds.
Total distance traveled = 18 + 6 + 2 + 2/3 ≈ 34.67 meters.
Hence, the total distance the ball has traveled before coming to rest is approximately 34.67 meters.