Chapter 4.1, Problem 11E

To determine

To calculate: The velocity of the object.

Answer to Problem 11E

The velocity of the object is $\frac{4}{\pi }\cos \left(3t\right)-\frac{4}{\pi }\sin \left(5t\right)$

Explanation of Solution

Given information: An object is moving along x- axis and its position is given by,

  $x(t)=s(t)\text{for}t\ge 0$ .

  $s(t)=\frac{4}{3\pi }\sin 3t+\frac{4}{5\pi }\cos 5t$

Formula used:

  $v=\frac{dx}{dt}$  ...... (1)

Where, v = velocity of object

x = position of object at time t

Calculation: Using the formula given in equation (1),

  $\begin{array}{l}v=\frac{dx}{dt}\\ \text{since}x(t)=s(t)\\ v=\frac{ds}{dt}\\ v=\frac{d\left(\frac{4}{3\pi }\sin 3t+\frac{4}{5\pi }\cos 5t\right)}{dt}\\ v=\frac{d\left(\frac{4}{3\pi }\sin 3t\right)}{dt}+\frac{d\left(\frac{4}{5\pi }\cos 5t\right)}{dt}\\ v=\frac{4}{3\pi }\cos \left(3t\right)\times \frac{d\left(3t\right)}{dt}+\frac{4}{5\pi }\left(-\sin \left(5t\right)\right)\times \frac{d\left(5t\right)}{dt}\\ v=\frac{4}{3\pi }\cos \left(3t\right)\times 3+\frac{4}{5\pi }\left(-\sin \left(5t\right)\right)\times 5\\ v=\frac{4}{\pi }\cos \left(3t\right)-\frac{4}{\pi }\sin \left(5t\right)\end{array}$

So the velocity of the object as a function of t is $\frac{4}{\pi }\cos \left(3t\right)-\frac{4}{\pi }\sin \left(5t\right)$