The Bouncing Effect
A ball is dropped from a height of 1 metre. It continues to bounce back up half its previous height continually till it comes to a stop. Calculate the total distance in metres covered by the ball.
NB: The total vertical distance covered by the ball after the several bounces is what we are looking for.
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1 solution
The ball is first dropped from a height of 1 metre
It bounces back up 1 / 2 a metre, goes down the same 1/2 a metre
again it bounces back up 1 / 4 of a metre, goes down the same 1 / 4 of a metre
then again it bounces back up 1 / 8 of a metre, goes down the same 1 / 8 of a metre
It continues this way infinitely, therefore we are looking at a sum to infinity
It happens to be a geometric progression if we exclude the first 1 metre
covered. Using the sum to infinity formula, we have,
a / ( 1 − r ) , where a is the first term and r is the common ratio
( 1 / 2 ) / ( 1 − 1 / 2 )
( 1 / 2 ) / ( 1 / 2 ) =1, but since there is a repetition of distances we have twice as much that is, 2
adding the 1 metre excluded at the beginning from the sequence, we have a total distance of 3 metres.