Question

A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the x-axis. Its kinetic energy K changes with time as dK/dt = $\gamma t,$ where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

• A. The force applied on the particle is constant
• B. The speed of the particle is proportional to time
• C. The distance of the particle from the origin increases linearly with time
• D. The force is conservative

Answer 1

Answer: (A), (B), (D)

The force is conservative

$\frac{dk}{dt}=\gamma t$ and $k=\frac{1}{2}m{v}^2$
Differentiating $k=\frac{1}{2}m{v}^2$ w.r.t. time
$\therefore \frac{dk}{dt}=mv\frac{dv}{dt}=\gamma t$
$\therefore m{\int }_0^{v}vdv=\gamma {\int }_0^{t}tdt$
$\frac{m{v}^2}{2}=\frac{\gamma t^2}{2}$
$v=\sqrt{\frac{\gamma }{m}}t$
$\Rightarrow v\propto t$; So option (B) is correct
since $F=ma$
And $a=\frac{dv}{dt}=\sqrt{\frac{\gamma }{m}}$
$\therefore F=m\sqrt{\frac{\gamma }{m}}=\sqrt{\gamma m}$ = constant
Here resultant force which is applied force in this case is constant.
So option (A) is correct
Since force applied is constant and displacement between any two points on x-axis will also be constant, thus work done will be independent of path , hence force is conservative in nature, So option D is correct
Option (A), (B) and (D) are correct answers