. A rubber ball is dropped from a certain height. After striking the floor, the ball bounces to 3/4th of the height it was dropped from. If the ball is initially dropped from a heigh of 100 meters, what is the total distance travelled by the ball before it comes to rest?
Question
. A rubber ball is dropped from a certain height. After striking the floor, the ball bounces to 3/4th of the height it was dropped from. If the ball is initially dropped from a heigh of 100 meters, what is the total distance travelled by the ball before it comes to rest?
Solution
The total distance travelled by the ball can be calculated by summing the distances of each bounce.
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The ball is initially dropped from a height of 100 meters, so the first distance is 100 meters.
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After the first bounce, the ball reaches a height of 3/4 * 100 = 75 meters. Since the ball travels this distance twice (up and then down), the total distance for the second bounce is 2 * 75 = 150 meters.
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For each subsequent bounce, the height reached by the ball is 3/4 of the height reached in the previous bounce, and the ball travels this distance twice.
This forms a geometric series with the first term a = 100 (the initial drop) and the common ratio r = 3/2 (since each subsequent term is 3/2 of the previous term due to the ball bouncing up and down).
The sum S of an infinite geometric series with first term a and common ratio r (where |r| < 1) is given by the formula S = a / (1 - r).
Substituting the given values into this formula gives S = 100 / (1 - 3/2) = 400 meters.
So, the total distance travelled by the ball before it comes to rest is 400 meters.
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