Chapter 4.3, Problem 10E

To determine

To calculate:The velocity of the object.

Answer to Problem 10E

The velocity of the object is $\frac{1}{8\sqrt{3}}$ .

Explanation of Solution

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

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

  $t=4$

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),

  $\begin{array}{l}v=\frac{dx}{dt}\\ v=\frac{d\left({\sin }^{-1}\left(\frac{\sqrt{t}}{4}\right)\right)}{dt}\\ v=\frac{1}{\sqrt{\left(1-{\left(\frac{\sqrt{t}}{4}\right)}^2\right)}}\times \frac{d\left(\frac{\sqrt{t}}{4}\right)}{dt}\\ v=\frac{1}{4\times \frac{\sqrt{\left(16-t\right)}}{4}}\times \frac{1}{2\sqrt{t}}\\ v=\frac{1}{2\sqrt{t}\times \sqrt{\left(16-t\right)}}\\ \text{Put}t=4,\\ v=\frac{1}{2\sqrt{4}\times \sqrt{\left(16-4\right)}}\\ v=\frac{1}{8\sqrt{3}}\end{array}$

So the velocity of the object is $\frac{1}{8\sqrt{3}}$ .