Multimedia

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  Example  2
A ball is dropped off the roof of a \(10 \rm \thinspace m\) high building. With each bounce, the ball regains \(75 \%\) of its height.

  1. Draw a diagram of the ball bouncing.  Stop when the ball hits the ground for the sixth time.


  2. Write out the vertical distance travelled by the ball with each bounce, until the ball hits the ground for the sixth time.  Recall that the ball must travel up and down with each bounce.

    First bounce = \(10 \rm \thinspace m \)
    Second bounce = \(2(10)(0.75) = 15 \)
    Third bounce = \(2(10)(0.75)^2 = 11.25 \)
    Fourth bounce = \(2(10)(0.75)^3 = 8.437\thinspace 5 \)
    Fifth bounce = \(2(10)(0.75)^4 = 6.328\thinspace 125 \)
    Sixth bounce = \(2(10)(0.75)^5 = 4.746 \thinspace ... \)

    Notice that by ignoring the first bounce, the rest of the sequence follows a geometric pattern, where the common ratio between pairs of consecutive terms is \(0.75\).

  3. Rounded to the nearest hundredth, what is the total vertical distance travelled by the ball when it hits the ground the sixth time? (Do not forget the distance of the first drop!)

    For the time being, ignore the first drop because it only travels down, not up and down. Consider the next term to be the first term, so use \(n = 5 \). Find the sum of the first five terms of the series, and then add the first drop to that total.

    \(\begin{align}
    t_1 &= 15 \\
    r &= 0.75 \\
    n &= 5 \\
    S_5 &= ? \\
    \end{align}\)
    \(\begin{align}
     S_n &= \frac{{t_1 \left( {r^n - 1} \right)}}{{r - 1}} \\
     S_5 &= \frac{{15\left[ {\left( {0.75} \right)^5 - 1} \right]}}{{0.75 - 1}} \\
     S_5 &= 45.761... \\
     S_5 &\buildrel\textstyle.\over = \thinspace 45.76 \\
     \end{align} \)

    Rounded to the nearest hundredth, the total vertical distance travelled by the ball is approximately \(10 \rm \thinspace m + 45.76 \rm \thinspace m \buildrel\textstyle.\over = \thinspace 55.76 \rm \thinspace m\).

  4. If the ball were to bounce infinitely many times, how far would it travel vertically?

    Ignore the first drop again. Because \(-1 < r < 1 \), you can use the formula for the sum of an infinite geometric series. Add the first drop after you calculate the answer.

    \(\begin{align}
     S_\infty &= \frac{{t_1 }}{{1 - r}} \\
     S_\infty &= \frac{{15}}{{1 - 0.75}} \\
     S_\infty &= 60 \\
     \end{align} \)
    Bouncing many times, the vertical distance travelled by the ball would be \(10 \rm \thinspace m + 60 \rm \thinspace m = 70 \rm \thinspace m\).