A ball is thrown 16 m into the air. The ball falls, rebounds to half of its previous height , and falls again... If the ball continues to rebounds and fall in this manner, find the total distance...

A ball is thrown 16 m into the air. The ball falls, rebounds to half of its previous height , and falls again.

.. If the ball continues to rebounds and fall in this manner, find the total distance the ball travels until it hits the ground for the sixth time?

Asked on by islnds

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tiburtius | High School Teacher | (Level 2) Educator

Posted on

The ball travels 16m up and 16m down until it hits the ground for the first time, then it travels 8m up and down and hits the ground for the second time etc. So you have 

 32+16+8+4+2+1=63m

This is easy to calculate for 6 bounces of the ball but if we needed to calculate distance traveled until e.g. 60th bounce we would have to use different approach.

Notice that numbers 32, 16, 8, 4, ... make geometric sequence

`a_n=32(1/2)^(n-1)`

and its sum makes geometric series.

Sum of first `n` terms of geometric sequence `a_n=a_1q^(n-1)` is equal to

`S_n=a_1 cdot (1-q^n)/(1-q)`.

Thus for `n=6` we have

 `S_6=32 cdot (1-(1/2)^6)/(1-1/2)=32cdot (63/64)/(1/2)=63`

`S_(60)=32cdot(1-(1/2)^(60))/(1-1/2)=2^5cdot((2^(60)-1)/2^(60))/(1/2)=2^5cdot(2^(60)-1)/(2^59)=(2^(60)-1)/(2^(54))`

`approx63.999999999999999944`

This series is also convergent and its limit is, as you can guess from above number, 64. 

 

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