A ball is droped from a height of 16 m on each bounce, it rebounds 0.5 of the distance it fell how far has the ball traveled. When it hits the ground for the fifth term?
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You probably meant:
How far has the ball traveled when it hits the ground for the fifth time?
The answer is
16 + 2*8 + 2*4 + 2*2 + 2*1 = 46 m
To find the total distance traveled by the ball after an arbitrary number of bounces, we can use the formula for the sum of a geometric series. The formula is given as:
S = a * (1 - r^n) / (1 - r),
where:
S = total distance traveled,
a = initial distance (16 m),
r = rebound factor (0.5),
n = number of bounces.
In this case, we want to find the distance traveled when the ball hits the ground for the fifth time. This means the ball has bounced four times, as the initial drop is not considered a bounce. Therefore, we substitute n = 4 into the formula:
S = 16 * (1 - 0.5^4) / (1 - 0.5).
Simplifying the equation:
S = 16 * (1 - 0.0625) / 0.5,
S = 16 * 0.9375 / 0.5,
S = 15 / 0.5,
S = 30.
Therefore, the ball has traveled a total distance of 30 meters when it hits the ground for the fifth time.