An object of mass 1000 g experiences a time dependent force $$\vec{F} = (2t\hat{i} + 3t^2\hat{j})$$ N. The power generated by the force at time t is :

Mass of the object, $$m = 1000\text{ g} = 1\text{ kg}$$.

Given time-dependent force
$$\vec{F}(t) = 2t\,\hat{i} + 3t^{2}\,\hat{j}\; \text{N}$$

Step 1 : Find acceleration.
Newton’s second law states $$\vec{F} = m\vec{a}$$, hence
$$\vec{a}(t) = \frac{\vec{F}}{m} = 2t\,\hat{i} + 3t^{2}\,\hat{j}$$ $$-(1)$$

Step 2 : Integrate acceleration to obtain velocity.
Assume the object starts from rest at $$t = 0$$, so $$\vec{v}(0) = 0$$.
Component-wise integration of $$(1)$$ gives

$$v_x(t) = \int 2t\,dt = t^{2} + C_x$$
$$v_y(t) = \int 3t^{2}\,dt = t^{3} + C_y$$

Using $$\vec{v}(0)=0$$ implies $$C_x = 0$$ and $$C_y = 0$$.
Therefore
$$\vec{v}(t) = t^{2}\,\hat{i} + t^{3}\,\hat{j}$$ $$-(2)$$

Step 3 : Calculate instantaneous power.
Power delivered by a force is the scalar (dot) product of force and velocity:
$$P(t) = \vec{F}\!\cdot\!\vec{v}$$

Using $$\vec{F}(t)$$ and $$\vec{v}(t)$$ from $$(2)$$:

$$\begin{aligned} P(t) &= (2t\,\hat{i} + 3t^{2}\,\hat{j}) \cdot (t^{2}\,\hat{i} + t^{3}\,\hat{j}) \\ &= 2t \cdot t^{2} + 3t^{2} \cdot t^{3} \\ &= 2t^{3} + 3t^{5} \end{aligned}$$

Step 4 : State the result.
The power generated by the force at time $$t$$ is
$$P(t) = 2t^{3} + 3t^{5}\ \text{W}$$.

Hence, the correct choice is Option D.