Power Produced by Time Dependent Force on Moving Body

Given the force $\vec{F} = 2t\hat{i} + 6t^2\hat{j}$ acting on a body of mass $m = 2\text{ kg}$ starting from rest, the acceleration is $\vec{a}(t) = \frac{\vec{F}}{m} = t\hat{i} + 3t^2\hat{j}$. Integrating acceleration gives the velocity vector $\vec{v}(t) = \frac{t^2}{2}\hat{i} + t^3\hat{j}$. The instantaneous power $P(t) = \vec{F} \cdot \vec{v} = t^3 + 6t^5$. Evaluating at $t = 2\text{ s}$ yields $P(2) = 2^3 + 6(2^5) = 8 + 192 = 200\text{ W}$.