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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
$A,B,C$
$A,B,D$
$A,B$
$A,C$
Solution
$\frac{ dK }{ dt }=\gamma t$
$K =\frac{1}{2} mv ^2$
$\frac{ dK }{ dt }=\frac{1}{2} m \times 2 v \frac{ dv }{ dt }=\gamma t$
$v \frac{ dv }{ dt }=\frac{\gamma t }{ m }$
$\int vdv =\frac{\gamma}{ m } \int tdt$
$\frac{ v ^2}{2}=\frac{\gamma}{ m } \cdot \frac{ t ^2}{2}$
$v =\sqrt{\frac{\gamma}{ m }} \cdot t \text { (proportional to time) }$
$a =\frac{ dv }{ dt }=\sqrt{\frac{\gamma}{ m }}$
$F = ma =\sqrt{\gamma m } \text { (constant) }$
