Question

A body of mass travels in a straight line with a velocity $v=k{x}^{3/2}$ where $k$ is a constant. The work done in displacing the body from $x=0$ to $x$ is proportional to:

• A. $x^{1/2}$
• B. $x^2$
• C. $x^3$
• D. $x^{5/2}$

Answer 1

The correct option is C $x^3$

Given,

Velocity, $\frac{dx}{dt}=v=k{x}^{3/2}$

Acceleration,

$a=\frac{dv}{dt}=\frac{d\left(k{x}^{3/2}\right)}{dt}=\frac{1}{2}k{x}^{1/2}\frac{dx}{dt}=\frac{1}{2}k{x}^{1/2}v$

$a=\frac{1}{2}k{x}^{1/2}\left(k{x}^{3/2}\right)=\frac{1}{2}{k}^2{x}^2$

$workdone=force\times displacemet=Mass\times acceleration\times displacement$

$\Rightarrow dW=m.a.dx=m.\frac{1}{2}{k}^2{x}^{2}dx$

On integrating

$W=m\frac{1}{2}{k}^2\left(\frac{x^3}{3}\right)$

Hence, Work done is proportional to $x^3$