A ball is dropped from a height of 80ft.The elasticity of this ball is such that it rebounds three-fourths of the distance it has fallen. How high does the ball rebound on the fifth bounce? Find a formula for the height the ball rebounds on the nth bounce?
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To find the height the ball rebounds on the nth bounce, we can use a formula that relates the rebound height to the initial drop height.
Given that the ball rebounds three-fourths of the distance it has fallen, the rebound height can be calculated as a fraction of the previous drop distance. Let's denote the drop height as H and the rebound height as R.
If we drop the ball from a height of H, it falls to the ground and rebounds to a height of R. Since the ball rebounds three-fourths of the distance it has fallen, we can write the relationship as:
R = (3/4)H
After the first rebound, the ball falls from R and rebounds to a height of (3/4)R. Continuing this pattern, we can write the relationship between the rebound height on the nth bounce (Rn) and the rebound height on the (n-1)th bounce (Rn-1) as:
Rn = (3/4)Rn-1
Using this formula, we can calculate the height of each rebound and find the height the ball rebounds on the fifth bounce.
Let's start by calculating the rebound heights for the first five bounces:
First bounce: R1 = (3/4)H
Second bounce: R2 = (3/4)R1 = (3/4)((3/4)H) = (9/16)H
Third bounce: R3 = (3/4)R2 = (3/4)((9/16)H) = (27/64)H
Fourth bounce: R4 = (3/4)R3 = (3/4)((27/64)H) = (81/256)H
Fifth bounce: R5 = (3/4)R4 = (3/4)((81/256)H) = (243/1024)H
Therefore, the ball rebounds to a height of (243/1024)H on the fifth bounce.
In general, the formula for the height the ball rebounds on the nth bounce can be written as:
Rn = (3/4)Rn-1 = (3/4)^n * H
Where H is the initial drop height, and n is the bounce number.