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 correct options are
A The speed of the particle is proportional to time
B The force applied on the particle is constant
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
$\therefore F=m\sqrt{\frac{\gamma }{m}}=\sqrt{\gamma m}$ = constant

So option (A) is correct

Since force applied is constant, it is conservative in nature, So option D is correct

Option (A), (B) and (D) are correct answers.