Chapter 4.4, Problem 39E

a.

To determine

To find: To find the maximum speed and acceleration.

Answer to Problem 39E

The maximum speed is $10\pi \text{ cm/sec}$ and acceleration is $a=-10{\pi }^2\cos (\pi t){\text{ cm/sec}}^2$ .

Explanation of Solution

Given:

  

A small frictionless cart, attached to the wall by a spring is pulled $10$ cm from its rest position and released at time $t=0$ to roll back and forth for $4$ sec. Its position at time t is,

  $s=10\cos (\pi t)$

Where s,t are in cm and sec respectively.

The time period is $4$ sec.

Formula used:

Velocity $v(t)=\frac{ds}{dt}$

Speed $sp=\left|v(t)\right|$

Calculation:

Velocity is found by $v(t)=\frac{ds}{dt}$

Substituting $s=10\cos (\pi t)$ in the above equation

  $\begin{array}{l}v(t)=\frac{d}{dt}(10\cos (\pi t))\\ v(t)=-10\pi \sin (\pi t)\end{array}$

Speed is

  $\begin{array}{l}sp=\left|v(t)\right|\\ sp=\left|-10\pi \sin (\pi t)\right|\\ sp=10\pi \left|\sin (\pi t)\right|\end{array}$

The acceleration is given by $a(t)=\frac{d^{2}s}{d{t}^2}$

  $\begin{array}{l}a=\frac{d}{dt}(-10\pi \sin (\pi t))\\ a=-10{\pi }^2\cos \left(\pi t\right)\end{array}$

Speed is maximum when

  $\begin{array}{l}\sin (\pi t)=1\\ \pi t=\frac{n\pi }{2}\\ n(odd)=1,3,5,7,\mathrm{.....}\\ t=\frac{n}{2},\\ n(odd)=1,3,5,7,\mathrm{....}\\ t=\frac{1}{2},\frac{3}{2},\frac{5}{7},\frac{7}{2},\mathrm{......}\end{array}$

The speed is maximum at $t=\frac{1}{2},\frac{3}{2},\frac{5}{7},\frac{7}{2}$

The cart remain at rest

  $\begin{array}{l}s=10\cos \left(\pi t\right)\\ s=0\end{array}$

b.

To determine

To find: To find the maximum speed and acceleration.

Answer to Problem 39E

The maximum speed is zero and acceleration at $t=0,1,2,3$ and cart remains at $s=10cm$ .

Explanation of Solution

Given:

  

A small frictionless cart, attached to the wall by a spring is pulled $10$ cm from its rest position and released at time $t=0$ to roll back and forth for $4$ sec. Its position at time t is,

  $s=10\cos (\pi t)$

Where s,t are in cm and sec respectively.

The time period is $4$ sec.

Formula used:

Velocity $v(t)=\frac{ds}{dt}$

Speed $sp=\left|v(t)\right|$

Calculation:

Acceleration is maximum when

  $\begin{array}{l}(\pi t)=n\pi ,n=0,1,2,3,4\\ t=0,1,2,3,4\\ a=-10{\pi }^2\cos (\pi t)\\ a=10{\pi }^2\end{array}$

Acceleration becomes maximum at $t=0,1,2,3,4\sec$

At this time the cart remain at

  $\begin{array}{l}s=10\cos \left(\pi t\right)\\ s=10\end{array}$

Speed is Zero.