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: (D)

The force is conservative

Answer 2

$\frac{dk}{dt}= \gamma t as k = \frac{1}{2}mv^{2}$
$\therefore \frac{dk}{dt} = mv \frac{dv}{dt} = \gamma t$
$\therefore m \int^{v}_{0} vdv = \gamma \int^{t}_{0} tdt$
$\frac{mv^{2}}{2} = \frac{\gamma t^{2}}{2}$
$v = \sqrt{\frac{ \gamma}{m} } t$ .....(i)
$a = \frac{dv}{dt} = \sqrt{\frac{\gamma}{m} } =$ constant
since F = ma
$\therefore F = m \sqrt{\frac{\gamma}{m}} = \sqrt{\gamma m } =$ constant