Chapter 4.1, Problem 10E

To determine

To calculate: The velocity of the object.

Answer to Problem 10E

The velocity of the object is $4t\sin \left(\pi -4t\right)+\cos \left(\pi -4t\right)$

Explanation of Solution

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

  $x(t)=s(t)\text{for}t\ge 0$ .

  $s(t)=t\cos \left(\pi -4t\right)$

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),

  $\begin{array}{l}v=\frac{dx}{dt}\\ \text{since}x(t)=s(t)\\ v=\frac{ds}{dt}\\ v=\frac{dt\cos \left(\pi -4t\right)}{dt}\\ v=t\times \frac{d\cos \left(\pi -4t\right)}{dt}+\cos \left(\pi -4t\right)\times \frac{dt}{dt}\\ v=t\times -\sin \left(\pi -4t\right)\times \frac{d\left(\pi -4t\right)}{dt}+\cos \left(\pi -4t\right)\times 1\\ v=-t\sin \left(\pi -4t\right)\times \left(0-4\right)+\cos \left(\pi -4t\right)\\ v=4t\sin \left(\pi -4t\right)+\cos \left(\pi -4t\right)\end{array}$

So the velocity of the object as a function of t is $4t\sin \left(\pi -4t\right)+\cos \left(\pi -4t\right)$