Question

The ball in Exercise 95 takes the following times for each fall. $$ \begin{array}{ll} s_{1}=-16 t^{2}+16, & s_{1}=0 \text { if } t=1 \\ s_{2}=-16 t^{2}+16(0.81), & s_{2}=0 \text { if } t=0.9 \\ s_{3}=-16 t^{2}+16(0.81)^{2}, & s_{3}=0 \text { if } t=(0.9)^{2} \\ s_{4}=-16 t^{2}+16(0.81)^{3}, & s_{4}=0 \text { if } t=(0.9)^{3} \end{array} $$ \(\vdots\) $$ s_{n}=-16 t^{2}+16(0.81)^{n-1}, \quad s_{n}=0 \text { if } t=(0.9)^{n-1} $$ Beginning with \(s_{2}\), the ball takes the same amount of time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is given by \(t=1+2 \sum_{n=1}^{\infty}(0.9)^{n}\) Find this total time.

Short answer

The total time elapsed before the ball comes to rest is 21 seconds.

Step-by-step solution

Identify the first term, common ratio and the formula of the infinite geometric series

In any geometric series, the first term \(a\) represents the initial term of the series, while the common ratio \(r\) is the constant value with which each term increases or decreases. In this case, the first term is 1 and the common ratio is 0.9. The sum \(S\) of an infinite geometric series is given by the formula \(\[ S = \frac{a}{1 - r} \]\), where \(a\) represents the first term and \(r\) represents the common ratio.

Apply the infinite geometric series formula

Substituting the values of \(a\) and \(r\) into the formula, we get \(\[ S = \frac{1}{1 - 0.9} \]\). From this, we can find the value of the series \(\[ \sum_{n=1}^{\infty}(0.9)^{n} \]\) by solving the equation.

Solve the equation to find the sum of the infinite series

Solving the equation gives us \(\[ S = 10 \]\) as the sum of the infinite series. This is the time it takes for the ball to bounce up from the second bounce onwards.

Subtract the initial time and multiply by 2

From the problem statement, we know that we have to add the initial time of 1 second, and multiply the sum of the consecutive times (from 2nd bounce) by 2 \(\[ t = 1 + 2 \times \((0.9)^{n}\) \]\). Substituting \(\[ S = 10 \]\) into the equation gives us the total time to be \(\[ t = 1 + 2 \times 10 = 21\] seconds\).