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

Given: Mass $m = 1 \text{ kg}$ and force $\vec{F} = (t \hat{i} + 2t^2 \hat{j}) \text{ N}$. The power developed by the force is $P = \vec{F} \cdot \vec{v}$.

Applying Newton's second law, acceleration is $\vec{a} = \frac{\vec{F}}{m} = t \hat{i} + 2t^2 \hat{j} \text{ m/s}^2$. Since the body starts from rest, velocity is found by integrating acceleration: $\vec{v} = \int (t \hat{i} + 2t^2 \hat{j}) \, dt = \left(\frac{t^2}{2}\right) \hat{i} + \left(\frac{2t^3}{3}\right) \hat{j} \text{ m/s}$.

Power is then calculated as $P = \vec{F} \cdot \vec{v} = (t \hat{i} + 2t^2 \hat{j}) \cdot \left(\frac{t^2}{2} \hat{i} + \frac{2t^3}{3} \hat{j}\right) = \frac{t^3}{2} + \frac{4t^5}{3}$.

At $t = 3$, $P(3) = \frac{27}{2} + \frac{4 \times 243}{3} = 13.5 + 324 = 337.5 \text{ W}$.