Question

A body of mass $m$ starts from rest and is moving in one dimension under the action of a force $F$ such that the force $F$ is directly proportional to $t$. Then, kinetic energy acquired by the body in travelling a certain distance $d$ along a straight line will be:

• A. Directly proportional to $t^2$
• B. Independent of $t$
• C. Directly proportional to $t^4$
• D. Directly proportional to $t$

Answer 1

The correct option is C Directly proportional to $t^4$

Given, force is directly proportional to $t$.
i.e $F\propto t\Rightarrow F=kt$
Also, mass $=m$, initial velocity $u=0$
Acceleration $a=\frac{F}{m}=\frac{kt}{m}$

Distance travelled by the body is $d$, hence applying 2nd equation of motion:
$\therefore d=ut+\frac{1}{2}a{t}^2[u=0]$
$\Rightarrow d=0+\frac{1}{2}a{t}^2=\frac{1}{2}\frac{\left(kt\right)}{m}{t}^2$
$\Rightarrow d=\frac{k{t}^3}{2m}$

Work done by force $W=Fd\cos 0^{\circ }=F\times d$ (in a straight line)
$\Rightarrow W=kt\times \frac{k{t}^3}{2m}$
$\Rightarrow W=\frac{k^2{t}^4}{2m}$
From work energy theorem,
$W_{\text{ext}}=\Delta KE$
$\Rightarrow \Delta KE=\frac{k^2{t}^4}{2m}$
$\therefore \Delta KE\propto t^4$, where $k=$ a positive constant
Option (c) is correct.