Chapter 4.3, Problem 9E

To determine

To calculate:The velocity of the object.

Answer to Problem 9E

The velocity of the object is $\frac{1}{\sqrt{7}}$ .

Explanation of Solution

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

  $x(t)={\sin }^{-1}\left(\frac{t}{4}\right)$ .

  $t=3$

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),

  $\begin{array}{l}v=\frac{dx}{dt}\\ v=\frac{d\left({\sin }^{-1}\left(\frac{t}{4}\right)\right)}{dt}\\ v=\frac{1}{\sqrt{\left(1-{\left(\frac{t}{4}\right)}^2\right)}}\times \frac{d\left(\frac{t}{4}\right)}{dt}\\ v=\frac{1}{4\times \frac{\sqrt{\left(16-t^2\right)}}{4}}\\ v=\frac{1}{\sqrt{\left(16-t^2\right)}}\\ \text{Put}t=3,\\ v=\frac{1}{\sqrt{\left(16-3^2\right)}}\\ v=\frac{1}{\sqrt{7}}\end{array}$

So the velocity of the object is $\frac{1}{\sqrt{7}}$ .