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; (D) The force is conservative

(C) The distance of the particle from the origin increases linearly with time

JEE Advanced · 2018 · 20 May 2018 · Paper 2 · Work, Power & Energy

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

d K / d t = γ t

, where

γ

is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

Answer: (A), (B), (D)

Given

\frac{d K}{d t} = γ t

, integrating gives

K = \frac{γ t ^{2}}{2}

. Since

K = \frac{1}{2} m v^{2}

, we have

v = t \frac{γ}{m}

, so speed is proportional to time (B is correct). The acceleration is

\frac{d v}{d t} = \frac{γ}{m}

, which is constant, so force

F = m \frac{d v}{d t} = mγ

is constant (A is correct). A constant force is conservative (D is correct). For displacement,

x = \int v d t = \frac{1}{2} t^{2} \frac{γ}{m}

, which is quadratic in time, not linear (C is incorrect).

JEE Advanced · 2018 · 20 May 2018 · Paper 2 · One or more correct options