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

Given force $\overrightarrow{\mathrm{F}}=( 2\mathrm{t} \hat{\mathrm{i}} +3{\mathrm{t}}^2 \hat{\mathrm{j}})$ 

According to Newton's law of motion,

 $\mathrm{m}\frac{\mathrm{d}\overrightarrow{\mathrm{v}}}{\mathrm{dt}}=2\mathrm{t} \hat{\mathrm{i}} +3{\mathrm{t}}^2 \hat{\mathrm{j}} (\mathrm{m} = 1\mathrm{kg})$

$$\begin{gathered} \Rightarrow {\int }_0^{\mathrm{v}}\mathrm{d}\overrightarrow{\mathrm{v}} = {\int }_0^{\mathrm{t}} (2\mathrm{t} \hat{\mathrm{i}} +3{\mathrm{t}}^2 \hat{\mathrm{j}}) \mathrm{dt} \\ \Rightarrow \overrightarrow{\mathrm{v}} = {\mathrm{t}}^2 \hat{\mathrm{i}} + {\mathrm{t}}^3 \hat{\mathrm{j}} \\ \mathrm{Power} \mathrm{P} = \overrightarrow{\mathrm{F}} .\overrightarrow{ \mathrm{v}} = (2\mathrm{t} \hat{\mathrm{i}} +3{\mathrm{t}}^2 \hat{\mathrm{j}}) . ( {\mathrm{t}}^2 \hat{\mathrm{i}} + {\mathrm{t}}^3 \hat{\mathrm{j}}) \end{gathered}$$

= (2t³ + 3t⁵)W