Chapter 3.5, Problem 45E

To determine

True or false the velocity and speed of the particle are the same at $t=\frac{\pi }{4}$ .Justify your answer.

Answer to Problem 45E

False becausespeed is positive and velocity is negative.

Explanation of Solution

Given information:A spring s bobbing up and down on the end of a spring according to $s(t)=-3\sin t$ .

Calculation:Firstly we find the velocity.

We know velocity is derivative of $s(t)$ .

  $\begin{array}{l}s(t)=-3\sin t\\ s\text{'}(t)=-3\cos t\end{array}$

Put $t=\frac{\pi }{4}$ in $s\text{'}(t)$ we get:

  $\begin{array}{l}s\text{'}(t)=-3\cos \frac{\pi }{4}\\ \mathrm{u}\mathrm{sing}\mathrm{graphing}\mathrm{calculator}\\ s\text{'}(t)=-3\times (0.70710)=-2.1213\end{array}$

Value is negative.

Therefore it’s travelling downward.

Speed is absolute value of velocity. So it is always positive.

  $\mathrm{speed}=\left|\mathrm{velocity}\right|=2.12$ .

Here speed is positive and velocity is negative.

Therefore they are not same.