Find number of bounces the ball will make before it bounces less than a given number of meters?

Posted on 2012-09-03
Last Modified: 2012-09-04
Earlier following question was asked:

As part of a science experiment, you drop a ball from various heights and measure how high it bounces on the first bounce.  The results of six drops are given below.

Drop height (m)                           0.5      1.5       2          2.5       4           5
First bounce height (m)              0.38     1.15    1.44     1.90     2.88      3.85

How high will the ball bounce if you drop it from a height of 6 meters?

Is there a pattern here?  I couldn't find a pattern.

This is high school Algebra 2 problem.  It's not homework.

Follow up question is as follows:

To continue the experiment, you must find the number of bounces the ball will make before it bounces less than a given number of meters.  Your experiment shows the ball's bounce height is always the same percent of height from which it fell before the bounce.  Find the average percent that the ball bounces each time.

What do they mean by finding the number of bounces the ball will make before it bounces less than a given number of meters.?

Thank you!
Question by:naseeam
    LVL 19

    Assisted Solution

    I think they left out "from a specified height".  For example, if it is dropped from 6m, how many bounces before it bounces less than, say, 1m.

    The "find the average percent...." is the key.
    LVL 31

    Accepted Solution

    If after falling H meters the next bounce is kH, which can be shown from the data, (here k is about 0.74 from previous question).

                Start Height  = H
    1st  Bounce Height  = kH
    2nd Bounce Height  = k²H
    3rd Bounce Height  = k³H

    Then after n bounces the the height h is  

    h = H*k^n

    If h is the required height to drop below since n is an integer find smallest n s.t.

    H*k^n < h

    k^n  < h/H

    Taking log both sides gives

    ln (k^n)   <  ln(h/H)


    n*ln(k) < ln(h/H)

    n < ln(h/H)/ln(k)

    so now find smallest integer n such that

    n < lg(h/H)/lg (0.74)


    n = ceiling ( lg(h/H)/lg (0.74))                        

    (ceiling = next integer that is higher)

    eg H = 6m  h=3m  k=0.74

    n = ceiling ( ln(3/6)/ln (0.74))

       = ceiing(-0.6931/-0.3011)

       = ceiling (2.3011)

       = 3

    (I am assuming good faith on the part of the questioner in that this is not homework as school does not start till tomorrow)
    LVL 18

    Assisted Solution

    >>>How high will the ball bounce if you drop it from a height of 6 meters?

    plot a graph of their data with small crosses for the values (drop height on the bottom horizontal, or x axis - bounce height on the vertical y axis)  make the bottom drop height axis go up to 6

    then put a ruler on the page and draw the line that best fits the crosses you plotted, and continue that line up above the values you plotted.

    now for 6 on your drop height, read off the predicted value of bounce height by going upwards to the line, and reading accross

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