A particle of mass m is initially at rest at the origin. It is su… - Vidyadip

MCQ

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
$A,B,C$
✓
$A,B,D$
C
$A,B$
D
$A,C$

✓

Answer

Correct option: B.

$A,B,D$

b
$\frac{ dK }{ dt }=\gamma t$

$K =\frac{1}{2} mv ^2$

$\frac{ dK }{ dt }=\frac{1}{2} m \times 2 v \frac{ dv }{ dt }=\gamma t$

$v \frac{ dv }{ dt }=\frac{\gamma t }{ m }$

$\int vdv =\frac{\gamma}{ m } \int tdt$

$\frac{ v ^2}{2}=\frac{\gamma}{ m } \cdot \frac{ t ^2}{2}$

$v =\sqrt{\frac{\gamma}{ m }} \cdot t \text { (proportional to time) }$

$a =\frac{ dv }{ dt }=\sqrt{\frac{\gamma}{ m }}$

$F = ma =\sqrt{\gamma m } \text { (constant) }$

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