Question

Figure shows the variation of force acting on a body with time. Assuming the body to start from rest, the variation of its momentum with time is best represented by which plot?

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

From $0$ to $2s$, at any time $t$
Force acting on body, $F=10t$
$\Rightarrow a=\frac{F}{m}=\frac{10t}{m}$
Since acceleration, $a=\frac{dv}{dt}$
$\Rightarrow {\int }_0^{v}dv={\int }_0^t\frac{10t}{m}dt$
$\Rightarrow v=\frac{5t^2}{m}$
Momentum $p=mv=5t^2$
Also slope of $p-t$ curve, $\frac{dp}{dt}=10t$
Hence from $0-2\text{sec}$ slope of $p-t$ curve increases with time.

At $t=2s$, $p=5(2{)}^2=20kg{\text{ms}}^{-1}$
$\Rightarrow v=\frac{20}{m}\text{m/s}$

From $2$ to $4s$, $F=40-10t$
$\Rightarrow {\int }_{\frac{20}{m}}^{v^{\prime }}dv={\int }_2^t\frac{40-10t}{m}dt$
$\Rightarrow v^{\prime }=\frac{40t-40-5t^2}{m}$
$\Rightarrow p=m{v}^{\prime }=40t-40-5t^2$
At $t=4s$, $p=40kg{\text{ms}}^{-1}$
From $2-4\text{sec},$ slope of $p-t$ curve $\frac{dp}{dt}=40-10t$
Hence from $2-4\text{sec}$, slope of $p-t$ curve decreases with time.
Hence, the correct answer is option (c)