Question

A ball of mass $m=1\text{kg}$ has an initial velocity $10\text{m/s}$. A variable force $F=20t+30t^2\text{N}$ is applied on the ball in the direction of motion of the ball for $1$ second. Find the final momentum of the ball.

• A. $30\text{Ns}$
• B. $20\text{Ns}$
• C. $10\text{Ns}$
• D. $15\text{Ns}$

Answer 1

The correct option is A $30\text{Ns}$

Impulse, $J=\int Fdt...\left(i\right)$
From impulse-momentum theorem,
$J=m\Delta v=P_f-P_i...\left(ii\right)$
$\Rightarrow P_f-P_i=\int Fdt$
where $P_f$ is the final momentum and $P_i=m{v}_i=1\times 10=10\text{Ns}$ is the initial momentum.

Integrating eq. (i) with limits of integration from $t=0$ to $t=1\text{s}$
$J=\underset{0}{\overset{1}{\int }}[20t+30t^2]dt$
$={[10t^2+10t^3]}_0^1$
$=10\left[\right(1-0)+(1-0\left)\right]=20\text{Ns}$

From Eq. $\left(ii\right)$:
$P_f-10=20$
$\Rightarrow P_f=30\text{Ns}$