Chapter 2.5, Problem 11E

To determine

To calculate: The velocity and acceleration at time $t$ and description of motion of the particle.

Answer to Problem 11E

The velocity is $v=5\cos t$ and acceleration is $a=-5\sin t$ .

Explanation of Solution

Given information:

The function is. $s=5\sin t$

Concept used:

The formula used is $\frac{d}{dx}\left(\sin x\right)=\cos x$ , $\frac{d}{dx}\left(\cos x\right)=-\sin x$ .

Calculation:

The function is $s=5\sin t$ .

Differentiate the function with respect to time $t$ , to find the velocity.

  $\begin{array}{c}\frac{d}{dt}\left(5\sin t\right)=5\frac{d}{dt}\left(\sin t\right)\\ v=5\cos t\end{array}$

Differentiate the function $v=5\cos t$ with respect to time $t$ , to find the acceleration.

  $\begin{array}{c}\frac{d}{dt}\left(5\cos t\right)=5\frac{d}{dt}\left(\cos t\right)\\ a=-5\sin t\end{array}$

From the position function it is clear that the mean position is $0$ and oscillates between $5$ to $-5$ and the period of motion of weight is $2\pi$ .

The greatest velocity of the weight can be achieved when it is either $-1$ or $1$ for that the value of $t$ should be of the form $k\pi$ and the velocity is $0$ when $t$ is of the form $k\frac{\pi }{2},k\text{ is odd}$

The acceleration is greatest when $t$ is either $-1$ or $1$ for that the value of $t$ should of the form $k\frac{\pi }{2},k\text{ is odd}$ and $0$ when $t$ is of the form $k\pi$ .

Conclusion: So, the velocity is $v=5\cos t$ and acceleration is $a=-5\sin t$ and .