Discussion Forum

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 kinectic energy K changes with time as $\frac{\text{d}K}{\text{d}t}=\gamma t$  , where $\gamma $ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

Answer:

Explanation:

$K=\frac{1}{2}mv^{2}\Rightarrow\frac{\text{d}K}{\text{d}t}\Rightarrow mv \frac{\text{d}v}{\text{d}t}$

 Given, $\frac{\text{d}K}{\text{d}t}=\gamma t\Rightarrow mv\frac{\text{d}v}{\text{d}t}=\gamma t$

   $\Rightarrow$   $\int_{0}^{v} v dv=\int_{0}^{t} \frac{\gamma}{m}t dt\Rightarrow\frac{v^{2}}{2}\Rightarrow\frac{\gamma}{m}\frac{t^{2}}{2}$

  $\Rightarrow$    $v=\sqrt{\frac{\gamma}{m}}t \Rightarrow a\Rightarrow\frac{\text{d}v}{\text{d}t}\Rightarrow\sqrt{\frac{\gamma}{m}}$

 $\therefore$    $F=ma=\sqrt{\gamma m}= constant$

 $\therefore$    $V=\frac{\text{d}s}{\text{d}t}=\sqrt{\frac{\gamma}{m}}t\Rightarrow s\Rightarrow\sqrt{\frac{\gamma}{m}}\frac{t^{2}}{2}$

 Note : Force is constant. In the website of IIT, option (d) is given correct. In the opinion of author all constant forces are not necessarily conservative. For example : viscous force at terminal velocity is a constant force but it is not conservative.