Chapter 9.8, Problem 30PPS

To determine

A graph to represent the arc length after each swing

Answer to Problem 30PPS

The sequence is $16,11.2,7.84,5.488,\mathrm{...}$

The graph is below

  

Explanation of Solution

Given information:

  

On each swing of a pendulum, the arc length is 60% of the previous swing

Then $r=0.6$

And $a_1=24$

Concept Used:

The equation for nth term is $a_n=a_1{\left(r\right)}^{n-1}$

The ratio is given by the formula

  $r=\frac{a_2}{a_1}=\frac{a_3}{a_2}$

Calculation:

In order to find the sequence of each bounce of ball is as shown below,

  $\begin{array}{c}{a}_n=a_1{\left(r\right)}^{n-1}\\ a_1=24\\ r=0.6\\ \text{equation for nth term is}\\ a_n=24{\left(0.6\right)}^{n-1}\\ a_2=24{\left(0.6\right)}^{2-1}\\ =24{\left(0.6\right)}^1\\ a_2=24\times 0.6\\ a_2=14.4\\ a_3=24{\left(0.6\right)}^{3-1}\\ =24{\left(0.6\right)}^2\\ =24\times 0.36\\ =8.64\\ a_4=24{\left(0.6\right)}^{4-1}\\ =24{\left(0.6\right)}^3\\ =24\times 0.216\\ =5.184\end{array}$

The sequence is $24,14.4,8.64,5.184,\mathrm{...}$

The graph is below