Chapter 3.5, Problem 14E

a.

To determine

To find: The body’s velocity, speed, and acceleration at the time $t$ .

Answer to Problem 14E

Velocity is $4\sin t$ ,speed is $\left|4\sin t\right|$ and the acceleration is $4\cos t$ .

Explanation of Solution

Given information: A body is moving in simple harmonic motion with position $s=1-4\cos t$ .

Calculation: Velocity is derivative of distance.

  $s\text{'}=-4(-\sin t)=4\sin t$ .

Speed is absolute value of velocity.

So speed is $\left|\mathrm{velocity}\right|=\left|4\sin t\right|$ .

The acceleration is derivative of velocity.

Acceleration is $s\text{'}\text{'}=4(\cos t)=4\cos t$ .

Thus, velocity is $4\sin t$ ,speed is $\left|4\sin t\right|$ and the acceleration is $4\cos t$ .

b.

To determine

The body’s velocity, speed, and acceleration at the time $t=\frac{\pi }{4}$ .

Answer to Problem 14E

Velocity is is $2\sqrt{2}$ ,speed is $2\sqrt{2}$ and the acceleration is $2\sqrt{2}$ .

Explanation of Solution

Given information: A body is moving in simple harmonic motion with position $s=1-4\cos t$ .

Calculation: Velocity is derivative of distance.

  $s\text{'}=4\sin t$ .

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

  $\begin{array}{l}s\text{'}=4\sin \frac{\pi }{4}\\ s\text{'}=4(\frac{1}{\sqrt{2}})=\frac{4}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}\\ s\text{'}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\end{array}$

Speed is absolute value of velocity.

Put $t=\frac{\pi }{4}$ So speed is

  $\begin{array}{l}=\left|4\sin t\right|\\ =\left|4\sin \frac{\pi }{4}\right|\\ =\left|4\times \frac{1}{\sqrt{2}}\right|=\left|\frac{4}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}\right|\\ =\left|\frac{4\sqrt{2}}{2}\right|=2\sqrt{2}\end{array}$ .

The acceleration is derivative of velocity.

Acceleration is $s\text{'}\text{'}=4(\cos t)=4\cos t$ .

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

  $\begin{array}{l}\mathrm{s}\text{'}\text{'}=4\cos \frac{\pi }{4}=\frac{4}{\sqrt{2}}\\ s\text{'}\text{'}=\frac{4}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\end{array}$

Thus, velocity is $2\sqrt{2}$ ,speed is $2\sqrt{2}$ and the acceleration is $2\sqrt{2}$ .

c.

To determine

Describe the motion of the body.

Answer to Problem 14E

The period of $s=1-4\cos t$ is $2\pi$ .

Explanation of Solution

Given information: A body is moving in simple harmonic motion with position $s=1-4\cos t$ .

Calculation: A body is moving in simple harmonic motion with position $s=1-4\cos t$ .

Draw the graph of this equation:

  

We observe from the graph the body swing between $-3\mathrm{and}5$ . So period is $2\pi$ .