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 $2\text{s}$:
At any time $t$, $F=10t$
Acceleration, $\therefore a=\frac{F}{m}=\frac{10t}{m}$
Also $a=\frac{dv}{dt}$
$\Rightarrow \underset{0}{\overset{v}{\int }}dv=\underset{0}{\overset{t}{\int }}\frac{10t}{m}dt\Rightarrow v=\frac{5t^2}{m}$
Momentum: $P=mv=5t^2$
Now, at $t=2\text{s}$
$P=5(2{)}^2=20{\text{kg ms}}^{-1}$

From $2$ to $4\text{s}$:
Force, $F=40-10t$
So, velocity at $t=2\text{s}$
$v=\frac{20}{m}$
$\therefore \underset{20/m}{\overset{v}{\int }}dv=\underset{2}{\overset{t}{\int }}\frac{40-10t}{m}dt\Rightarrow v=\frac{1}{m}[40t-40-5t^2]$
$P=mv=40t-40-5t^2$
Therefore, at $t=4,P=40\times 4-40-5\times 4^2=40{\text{kg ms}}^{-1}$
Since $a=\frac{dv}{dt}=\frac{F}{m}$
As $F$ is decreasing from $2-4s$, therefore slope of $P-t$ curve will decrease from $2-4s$.
(c) Option is correct.