Question

At time $t=0$ a particle starts moving along the x axis. If its kinetic energy increases uniformly with $t$, the net force acting on it must be

• A. Constant
• B. Proportional to $t$
• C. Inversely proportional to $t^2$
• D. Proportional to $1/\sqrt{t}$

Answer 1

The correct option is D Proportional to $1/\sqrt{t}$

Linear dependency with initial KE as $zero$ is given by $KE=bt$, where $b$ is a proportionality constant.

as we know that $KE=\frac{1}{2}m{v}^2$ so we may write, $v^2=\frac{2bt}{m}=ct$ ........... (1)

where $c=\frac{2b}{m}$ is a new constant.

Now to find acceleration we need $a=\frac{d}{dt}v=\frac{d}{dt}\sqrt{ct}$, from equation -1

thus we get $a=\sqrt{c}\times \frac{1}{2\sqrt{t}}$

so force will be $F=ma=\frac{m\sqrt{c}}{2}\times \frac{1}{\sqrt{t}}$

or we can write $F\propto \frac{1}{\sqrt{t}}$