Chapter 11.2, Problem 7QR

To determine

To calculate: The velocity and acceleration of a particle moving along a line if its position at time $t$ is given by $x\left(t\right)=t\sin t$ .

Answer to Problem 7QR

The velocity of the particle is $\sin t+t\cos t$ and the acceleration is $2\cos t-t\sin t$ .

Explanation of Solution

Given information:

Position $x\left(t\right)=t\sin t$

Time $=t$

Formula used:

Velocity of an object with distance $x(t)$ at time $t$ :

  $v=\frac{d}{dt}x(t)$

Accelerationof an object with velocity $v$ at time $t$ :

  $a=\frac{d}{dt}v$

Calculation:

Given that the position of the particle at time is $x\left(t\right)=t\sin t$ .

Now, calculate velocity. Use the formula $v=\frac{d}{dt}x(t)$ .

  $\begin{array}{c}v=\frac{d}{dt}x(t)\\ =\frac{d}{dt}t\sin t\\ =\sin t+t\cos t\end{array}$

Similarly, calculate acceleration. Use the formula $a=\frac{d}{dt}v$ .

  $\begin{array}{c}a=\frac{d}{dt}v\\ =\frac{d}{dt}\left(\sin t+t\cos t\right)\\ =\cos t+\cos t-t\sin t\\ =2\cos t-t\sin t\end{array}$

Conclusion:

Hence, the required velocity of the particle is $\sin t+t\cos t$ and the acceleration is $2\cos t-t\sin t$ .