A rubber ball is dropped from a height of 10meters,suppose it rebounds one-half the distance after each fall.find the total distance the ball travels before coming to rest.
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To find the total distance the ball travels before coming to rest, we need to calculate the sum of all the distances the ball travels during its falls and rebounds.
Let's break down the scenario step by step:
1. Distance traveled during the first fall: The ball is dropped from a height of 10 meters, so during the first fall, it travels a distance of 10 meters.
2. Distance traveled during the first rebound: After the first fall, the ball rebounds half the distance it fell. So, it rebounds a distance of 10/2 = 5 meters.
3. Distance traveled during the second fall: The ball is dropped again from a height of 5 meters (distance traveled during the rebound). Therefore, during the second fall, it travels a distance of 5 meters.
4. Distance traveled during the second rebound: After the second fall, the ball rebounds half the distance it fell. So, it rebounds a distance of 5/2 = 2.5 meters.
5. Distance traveled during the third fall: The ball is dropped again from a height of 2.5 meters (distance traveled during the second rebound). Therefore, during the third fall, it travels a distance of 2.5 meters.
6. Distance traveled during the third rebound: After the third fall, the ball rebounds half the distance it fell. So, it rebounds a distance of 2.5/2 = 1.25 meters.
This pattern continues, with each rebound being half the distance of the previous fall until the ball eventually comes to rest.
To find the total distance traveled, we need to add up all the distances from the falls and rebounds.
Total Distance = Distance of first fall + Distance of first rebound + Distance of second fall + Distance of second rebound + ...
Total Distance = 10 + 5 + 5 + 2.5 + 2.5 + 1.25 + 1.25 + ...
Notice that the distances traveled during the falls and rebounds follow a geometric series, where the first term is 10 and the common ratio is 1/2.
Using the formula for the sum of an infinite geometric series, we can calculate the total distance:
Total Distance = 10 / (1 - 1/2)
Total Distance = 10 / (1/2)
Total Distance = 10 * 2
Total Distance = 20 meters
Therefore, the rubber ball travels a total distance of 20 meters before coming to rest.