A body of mass 1 kg begins to move under the action of a time dependent force $$\vec{F} = (t\hat{i} + 3t^2\hat{j})$$ 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 $$t = 2$$ s, will be _____ W.

A body of mass $$m = 1$$ kg starts from rest under the force $$\vec{F} = (t\hat{i} + 3t^2\hat{j})$$ N, so the acceleration is $$\vec{a} = \frac{\vec{F}}{m} = t\hat{i} + 3t^2\hat{j}$$ m/s².

Since the initial velocity is $$\vec{v}(0)=0$$, integrating the acceleration from 0 to $$t$$ gives the velocity as $$ \vec{v} = \int_0^t \vec{a}\,dt = \frac{t^2}{2}\hat{i} + t^3\hat{j}. $$

Next, at $$t = 2$$ s, we obtain $$ \vec{v}(2) = \frac{4}{2}\hat{i} + 8\hat{j} = 2\hat{i} + 8\hat{j} $$ m/s and $$ \vec{F}(2) = 2\hat{i} + 12\hat{j} $$ N, so the power is $$ P = \vec{F}\cdot\vec{v} = (2)(2) + (12)(8) = 4 + 96 = 100 \text{ W}. $$ Therefore the correct answer is 100 W.