Chapter 4, Problem 70RE

a.

To determine

Give parametric equation that can be used to simulate the motion of the particle.

Answer to Problem 70RE

  $x(t)=t,y(t)=10\cos (t+\frac{\pi }{4})$

Explanation of Solution

Given information: The position at time $t\ge 0$ of a particle moving along the s-axis is,

  $s(t)=10\cos (t+\frac{\pi }{4}).$

Calculation:

  $\begin{array}{l}s(t)=10\cos (t+\frac{\pi }{4}).\\ x(t)=t,y(t)=10\cos (t+\frac{\pi }{4}).\end{array}$

b.

To determine

What is the particle’s initial position (t=0)?

Answer to Problem 70RE

  $s(0)=5\sqrt{2}$

Explanation of Solution

Given information: The position at time $t\ge 0$ of a particle moving along the s-axis is,

  $s(t)=10\cos (t+\frac{\pi }{4}).$

Calculation:

  $\begin{array}{l}s(t)=10\cos (t+\frac{\pi }{4}).\\ s(0)=10\cos (0+\frac{\pi }{4})\\ s(0)=5\sqrt{2}\end{array}$

c.

To determine

What points reached by the particle are farthest to the left and right of the origin?

Answer to Problem 70RE

Furthest to the right when s=10 and furthest to the left when s=-10.

Explanation of Solution

Given information: The position at time $t\ge 0$ of a particle moving along the s-axis is,

  $s(t)=10\cos (t+\frac{\pi }{4}).$

Calculation:

  $\begin{array}{l}s(t)=10\cos (t+\frac{\pi }{4})=0.\\ s\text{'}(t)=10[-\sin (t+\frac{\pi }{4})]=-10\sin (t+\frac{\pi }{4})\\ s\text{'}(t)=0\\ -10\sin (t+\frac{\pi }{4})=0\\ \sin (t+\frac{\pi }{4})=0\\ (t+\frac{\pi }{4})=0,\pi ,2\pi \mathrm{.....}\\ t=-\frac{\pi }{4},\frac{3\pi }{4},\frac{7\pi }{4}\\ s(\frac{3\pi }{4})=10\cos (\frac{3\pi }{4}+\frac{\pi }{4})=10\cos \pi =10(-1)=-10\\ s(\frac{7\pi }{4})=10\cos (\frac{7\pi }{4}+\frac{\pi }{4})=10\cos 2\pi =10(1)=10\end{array}$

Furthest to the right when s=10 and furthest to the left when s=-10.

d.

To determine

When does the particle first reach the origin ?What are its velocity ,speed and acceleration then?

Answer to Problem 70RE

At $t=\frac{\pi }{4}$ velocity=-10,speed=10,acceleration=0.

Explanation of Solution

The position at time $t\ge 0$ of a particle moving along the s-axis is,

  $s(t)=10\cos (t+\frac{\pi }{4}).$

Calculation:

  $\begin{array}{l}s(t)=10\cos (t+\frac{\pi }{4})=0.\\ \cos (t+\frac{\pi }{4})=0\\ (t+\frac{\pi }{4})=\frac{\pi }{2}\\ t=\frac{\pi }{4}\\ s\text{'}(t)=-10\sin (t+\frac{\pi }{4})=-10\sin (\frac{\pi }{4}+\frac{\pi }{4})=-10\\ \left|s\text{'}(t)=\left|-10\right|\right|=10\\ s\text{'}\text{'}(t)-10\cos (t+\frac{\pi }{4})\\ s\text{'}\text{'}(\frac{\pi }{4})=-10\cos (\frac{\pi }{4}+\frac{\pi }{4})=-10\cos \frac{\pi }{2}=-10(0)=0\\ s\text{'}\text{'}(\frac{\pi }{4})=0\end{array}$

At $t=\frac{\pi }{4}$ velocity=-10,speed=10,acceleration=0.