Question

The velocity of a body of mass $2\text{kg}$ as a function of $t$ is given by $v\left(t\right)=2t\hat{l}+t^{2}j,$
Find the momentum and the force acting on it, at time $t=2s.$

Answer 1

Step 1 : Find velocity of the body at $t=2s.$
Given, velocity of the body
$\overrightarrow{v}\left(t\right)=2t\hat{l}+t^2\hat{j}$
So, velocity of the body at $t=2s$
$\overrightarrow{v}=2\times 2\hat{l}+(2{)}^2\hat{j}=(4\hat{l}+4\hat{j})$

Step 2: Find momentum of the body at $t=2s.$
Formula used: $\overrightarrow{p}=m\overrightarrow{v}$
Given, mass of the body $m=2\text{kg}$
So, momentum of the body, $\overrightarrow{p}=m\overrightarrow{v}$
$\overrightarrow{p}=2(4\hat{l}+4\hat{j})=(8\hat{i}+8\hat{j})\text{kg}m/s$

Step 3: Find acceleration of the body at $t=2s.$
Formula used: $\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}$
$\overrightarrow{a}=\frac{d}{dt}(2t\hat{l}+t^2\hat{j})$
$\overrightarrow{a}=2\hat{l}+2t\hat{j}$
Acceleration of the body at $t=2s$
$\overrightarrow{=}(2\hat{l}+4\hat{j})m/s^2$

Step 4: Find force acting on the body at $t=2s.$
Formula used: $\overrightarrow{F}=m\overrightarrow{a}$
Force on the body at $t=2s$
$\overrightarrow{F}=m\overrightarrow{a}$
$\overrightarrow{F}=2(2\hat{l}+4\hat{j})=(4\hat{l}+8\hat{j})N$

Final answer:
$(8\hat{l}+8\hat{j})\text{kg}m/s;(4\hat{l}+8\hat{j})N$