Question

A constant force acts on a body of mass $m$ at rest. The velocity $v$ acquired in traversing a specific distance depends on $m$ as

• A. $v\propto \frac{1}{m}$
• B. $v\propto \frac{1}{\sqrt{m}}$
• C. $v\propto m$
• D. $v\propto \sqrt{m}$

Answer 1

The correct option is B

$v\propto \frac{1}{\sqrt{m}}$

Explanation for the correct answer:

Option (B): $v\propto \frac{1}{\sqrt{m}}$

Step 1. By Newton’s second law, the formula for force

Force is equal to the product of the mass and the acceleration.

$F=ma$ [Where, $F$ is force, $m$ is mass, $a$ is acceleration

$\Rightarrow a=\frac{F}{m}$

$$\begin{gathered} \Rightarrow \frac{F}{m}=\frac{dv}{dt} \\ \Rightarrow \frac{F}{m}=\frac{dv}{dx}.\frac{dx}{dt} \\ \Rightarrow \frac{F}{m}=\frac{dv}{dx}·v \end{gathered}$$ $\left[\frac{dx}{dt}=v\right]$ [Where, $v$ is velocity, $x$ is distance]

$\Rightarrow \frac{Fdx}{m}=v·dv$

Integrating we get,

$\Rightarrow \int \frac{Fdx}{m}=\int v·dv$

$\Rightarrow \frac{Fx}{m}=\frac{v^2}{2}+c$ [$c$ is a constant]

Step 2. Solving the above equation by substituting$\mathbf{}\mathit{v}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$ at $\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$,

We get $c=0$

$$\begin{gathered} \Rightarrow \frac{Fx}{m}=\frac{v^2}{2} \\ \Rightarrow v=\sqrt{\frac{2Fx}{m}} \end{gathered}$$

so, $v\propto \frac{1}{\sqrt{m}}$

Hence, the correct option is $\left(B\right)$