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Math Help - find distance of bouncing ball

  1. #1
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    find distance of bouncing ball

    from height of 8 meters a ball fell down and each time it bounces half the distance back.what will be the distance travelled?
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  2. #2
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    Sum to infinity

    We use the formula for a sum to infinty:

    S=a/(1-r) where r is the constant multiple, 0.5 and a is the starting height 8

    Therefore S=8/(1-0.5)=16m total travelled

    This is the same as adding 8+8/2+8/4+8/8+8/16+8/32...8/2^n
    which is 8+4+2+1+0.5+0.25+0.125...which eventually reaches very close to our answer of 16.
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  3. #3
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    ok lets write out the first few terms of the distance it travels to see if we can see a pattern

    first it falls 8 then bounces back up to 4 then falls back down 4 the bounces back up 2 and falls back down 2 then bounces back up 1 and falls back down 1.......

    we can write this as
    <br />
D=8+4+4+2+2+1+1+\ldots

    grouping the the pairs of terms we have
    <br />
D=8+8+4+2+\ldots

    you should notice that from the second term onwards forms a geometric progression with first term 8 and common ratio 0.5 so we can write it as

    <br />
D=8+\sum_{i=1}^\infty 8(0.5)^{i-1}

    you should know that for common ratio r, first term a and |r|<1 this infinite sum has a value of \frac{a}{1-r}

    so
    <br />
D=8+\frac{8}{1-0.5}=24m
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  4. #4
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    hiiii
    thanks for ur ans
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  5. #5
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    thanks

    thanks for ur ans
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