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)

$\frac{dk}{dt}=\gamma task=\frac{1}{2}m{v}^2$
$\therefore \frac{dk}{dt}=mv\frac{dv}{dt}=\gamma t$
$\therefore m{\int }_0^{v}vdv=\gamma {\int }_0^{t}tdt$
$\frac{m{v}^2}{2}=\frac{\gamma t^2}{2}$
$v=\sqrt{\frac{\gamma }{m}}t$ .....(i)
$a=\frac{dv}{dt}=\sqrt{\frac{\gamma }{m}}=$ constant
since F = ma
$\therefore F=m\sqrt{\frac{\gamma }{m}}=\sqrt{\gamma m}=$ constant