A body of mass $1kg$ begins to move under the action of a time dependent force $\overrightarrow{\mathrm{F}}=(2\mathrm{t} \hat{\mathrm{i}}+3{\mathrm{t}}^2 \hat{\mathrm{j}}) \mathrm{N},$ where $\hat{i} and \hat{j}$ are unit vectors along $x-$axis and $y-$axis respectively. What power will be developed by the force at the time $t$?

$$\begin{gathered} \mathrm{Here}, \overrightarrow{\mathrm{F}}=\left(2\mathrm{t} \hat{\mathrm{i}}+3{\mathrm{t}}^2 \hat{\mathrm{j}}\right)\mathrm{N}, \mathrm{m}=1 \mathrm{kg} \\ \Rightarrow \overrightarrow{\mathrm{a}}=\frac{\overrightarrow{\mathrm{F}}}{\mathrm{m}}=\frac{\left(2\mathrm{t} \hat{\mathrm{i}}+3{\mathrm{t}}^2 \hat{\mathrm{j}}\right) \mathrm{N}}{1 \mathrm{kg}}=\left(2\mathrm{t} \hat{\mathrm{i}}+3{\mathrm{t}}^2 \hat{\mathrm{j}}\right) {\mathrm{ms}}^{-2} \end{gathered}$$

$\text{ Velocity of the body at time }t \mathrm{is} \mathrm{given} \mathrm{by}:$

$\overrightarrow{\mathrm{v}}=\int \overrightarrow{\mathrm{a}}\mathrm{dt}=\int \left(2\mathrm{t} \hat{\mathrm{i}}+3{\mathrm{t}}^2 \hat{\mathrm{j}}\right)\mathrm{dt}=\left({\mathrm{t}}^2 \hat{\mathrm{i}}+{\mathrm{t}}^3 \hat{\mathrm{j}}\right) {\mathrm{ms}}^{-1}$

$\therefore \text{ Power developed by the force at time }t,$

$\mathrm{P}=\overrightarrow{\mathrm{F}}·\overrightarrow{\mathrm{v}}=\left(2\mathrm{t}\hat{\mathrm{i}}+3{\mathrm{t}}^2\hat{\mathrm{j}}\right)·\left({\mathrm{t}}^2\hat{\mathrm{i}}+{\mathrm{t}}^3\hat{\mathrm{j}}\right) \mathrm{W}=\left(2{\mathrm{t}}^3+3{\mathrm{t}}^5\right) \mathrm{W}$