Question

If the force acting on a body is inversely proportional to its speed, then its Kinetic energy is:
A. Linearly related to time
B. Inversely proportional to time
C. Inversely proportional to the square of time
D. A constant

Answer 1

Hint: In order to solve this question, we will first formulate a differential equation in terms of speed and time according to the given information and then we will solve that equation to find out the relation between kinetic energy and time.

Complete step by step answer:
According to the question, we have given that
$F \propto \dfrac{1}{v}$
where $F, v$ be the force and velocity of the body.
$ \Rightarrow F = \dfrac{k}{v} \to (i)$
where $k$ is a proportionality constant.
Now as we know, force can be written as derivative of velocity with time as,
$F = m\dfrac{{dv}}{{dt}}$ where m is the mass of the body,

Put this value of force in equation (i) we get,
$m\dfrac{{dv}}{{dt}} = \dfrac{k}{v}$
Rearranging the differential terms
$vdv = \dfrac{k}{m}dt$
Integrating both sides we get,
$\int {vdv} = \int {\dfrac{k}{m}dt} $
Using rules of integrations as $\int {xdx = \dfrac{{{x^2}}}{2};\int {dx = x} } $
And we know $k, m$ are the constant terms so, we get
$\dfrac{{{v^2}}}{2} = \dfrac{k}{m}t + C$
where $C$ is the integration constant.

Now multiply the whole equation with m we get,
$\dfrac{{m{v^2}}}{2} = kt + Cm$
Now, we know that $K.E = \dfrac{1}{2}m{v^2}$ so from equation $\dfrac{{m{v^2}}}{2} = kt + Cm$ we can write it as
$K.E = kt + Cm$
where K.E stands for Kinetic energy so,
$\therefore K.E \propto t$
Kinetic energy is directly proportional to time linearly.

Hence, the correct option is A..

Note: It should be remembered that, whenever we solve an indefinite integral equation, we have to add an arbitrary constant and here the product of constant $C$ and mass $m$ can also be written as another constant that’s why, we get direct relation between Kinetic energy and time.