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Math Help - Ball drop and rebound question =/?

  1. #1
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    Ball drop and rebound question =/?

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    Last edited by SarahGr; May 14th 2009 at 07:29 PM.
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  2. #2
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    This is an infinite geometric series question with r=.8 and the coefficient equal to 12. Since r<1, this series converges and the ball actually travels a finite distance.

    \Sigma_{k=0}^{k=\infty}r^k=\frac{1}{1-r} for |r|<1 that formula holds true, but in our case here, \Sigma_{k=0}^{k=\infty}12(r)^k=\frac{12}{1-r} however I forgot that the ball bounces and falls so Soroban is correct
    Last edited by artvandalay11; May 13th 2009 at 08:57 AM.
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  3. #3
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    Hello, Sarah!

    A ball is dropped from a height of 12 feet.
    The ball hits the ground and rebounds 80% of the distance it falls each time.
    In theory, how far will the ball travel before coming to rest?
    Let's baby-step our way through this . . .

    It falls 12 feet.

    \text{Then it bounces up }80\%\text{ of }12\:=\:(0.8)12\text{ feet.}
    . . Then it falls that distance.

    \text{Then it bounces up }80\%\text{ of }(0.8)12 \:=\:(0.8)^212\text{ feet.}
    . . Then it falls that distance.

    \text{Then it bounces up }80\%\text{ of }(0.8)^212 \:=\:(0.8)^312\text{ feet.}
    . . Then it falls that distance.
    And so on . . .


    We can see that the total distance is:

    . . D \;=\;12 + 2(0.8)12 + 2(0.8)^212 + 2(0.8)^312 + 2(0.8)^412 + \hdots

    . . D \;=\;12 + 2(0.8)12\underbrace{\bigg[1 + 0.8 + 0.8^2 + 0.8^3 + \hdots \bigg]}_{\text{geometric series}}

    The geometric series has first term a = 1, common ratio r = 0.8
    . . Its sum is: . \frac{1}{1-0.8} \:=\:\frac{1}{0.2} \:=\:5


    Therefore: . D \;=\;12 + 2(0.8)12(5) \;=\;108\text{ feet}

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    Last edited by SarahGr; May 14th 2009 at 07:29 PM.
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