Work Power and Energy 5 Question 10
13. 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 $d K / d t=\gamma t$, where $Y$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?
(2018 Adv.)
(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
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Answer:
Correct Answer: 13. (a, b)
Solution:
- $(a, b)$ $\quad K=\frac{1}{2} m v^{2}$
$\Rightarrow \quad \frac{d K}{d t}=m v \frac{d v}{d t}$
Given, $\quad \frac{d K}{d t}=\gamma t$
$\Rightarrow \quad m v \frac{d v}{d t}=\gamma t$
$\Rightarrow \quad \int _0^{v} v d v=\int _0^{t} \frac{Y}{m} t d t$
$\Rightarrow \quad \frac{v^{2}}{2}=\frac{Y}{m} \frac{t^{2}}{2}$
$\Rightarrow \quad v=\sqrt{\frac{Y}{m}} t$
$\Rightarrow \quad a=\frac{d v}{d t}=\sqrt{\frac{Y}{m}}$
$\therefore \quad F=m a=\sqrt{Y m}=$ constant
$\therefore \quad V=\frac{d s}{d t}=\sqrt{\frac{Y}{m}} t \Rightarrow s=\sqrt{\frac{Y}{m}} \frac{t^{2}}{2}$
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.
