Question:
The velocity of a body of mass 2 kg as a function of t is given by
$v(t)=2 t \hat{i}+t^{2} \hat{j}$ Find the momentum and the force acting on it at time $t=2$ sec.
Solution:
m = 2 kg
$\vec{v}(t)=2 t \hat{i}+t^{2} \hat{j}$
$\vec{v}$ at $2 \mathrm{sec}, \vec{v}(2 t)=2(2 t) \hat{i} 2^{2} \hat{j}, v(2)=4 \hat{i}+4 \hat{j}$
Momentum, $\underset{P}{\rightarrow}(2)=m \vec{v}(2)$
$\vec{P}(2)=2[4 \hat{i}+4 \hat{j}], p(2)=8 \hat{i}+8 \hat{j} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$
$\vec{F}=m \vec{a}$
$\vec{v}(t)=2 t \hat{i}+t^{2} \hat{j}$
$\vec{a}(t)=\frac{d \vec{v}(t)}{d t}=2 \hat{i}+2 t \hat{j}$
$\vec{F}(2)=2(2 \hat{i}+4 \hat{j})=4 \hat{i}+8 \hat{j} N$