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 = t, where is a positive constant of appropriate dimensions. Which of the following statements is (are) true? [JEE Adv 2018 P2]

The force applied on the particle is constant

The speed of the particle is proportional to time

The distance of the particle from the origin increases linearly with time

Test Index

Work, Power & Energy

Section: Physics

Question : 4 of 22

Marks: +1, -0

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

\gamma

Solution:

\frac{dk}{dt} = \gamma t

k = \frac{1}{2} m v^2 \therefore \frac{dk}{dt} = m v \frac{dv}{dt} = \gamma t \therefore m \int\limits_{0}^{v} v \, dv = \gamma \int\limits_{0}^{t} t \, dt \;\;\;\; \frac{m v^2}{2} = \frac{\gamma t^2}{2}

v = \sqrt{\frac{\gamma}{m}} t \;\;\;\; a = \frac{dv}{dt} = \sqrt{\frac{\gamma}{m}} =

Constant Since

F = m a

\therefore F = m \sqrt{\frac{\gamma}{m}} = \sqrt{\gamma m} =

Constant

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