Question

A body of mass $1kg$ begins to move under the action of a time dependent force $\vec{F}=\left(\hat{i}+3t^2\hat{j}\right)N$, where $\hat{i}$ and $\hat{j}$ are the unit vectors along $x$ and $y$ axis. The power developed by above force, at the time $t=2s$, will be _________W.

Answer 1

Answer: 100

$\vec{F}=t\hat{i}+3t^2\hat{j}$
$\frac{md\vec{v}}{dt}=t\hat{i}+3t^2\hat{j}$
$m=1kg,\underset{0}{\overset{\vec{v}}{\int }}dv=\underset{0}{\overset{t}{\int }}tdt\hat{i}+\underset{0}{\overset{t}{\int }}3t^{2}dt\hat{j}$
$\vec{v}=\frac{t^2}{2}\hat{i}+t^3\hat{j}$
Power $=\vec{F}\cdot \vec{V}=\frac{t^3}{2}+3t^5$
At $t=2$, power $=\frac{8}{2}+3\times 32$
$=100$