Given Information:

Force: $\vec{F} = (2t \hat{i} + 3t^2 \hat{j}) \, \text{N}$

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

Determine Acceleration:

Using Newton's second law, $\overrightarrow{F} = m \overrightarrow{a}$.

Thus, the acceleration, $\overrightarrow{a} = 2t \hat{i} + 3t^2 \hat{j}$.

Velocity Calculation:

To find velocity, integrate acceleration with respect to time:

$ \frac{d\overrightarrow{v}}{dt} = 2t \hat{i} + 3t^2 \hat{j} $

Integrate to find $\overrightarrow{v}$:

$ \overrightarrow{v} = \int (2t \hat{i} + 3t^2 \hat{j}) \, dt = t^2 \hat{i} + t^3 \hat{j} + \vec{C} $

(Assuming initial velocity is zero, $\vec{C} = 0$, hence $\overrightarrow{v} = t^2 \hat{i} + t^3 \hat{j}$).

Calculate Power:

Power is defined as the dot product of force and velocity vectors:

$ P = \overrightarrow{F} \cdot \overrightarrow{v} = (2t \hat{i} + 3t^2 \hat{j}) \cdot (t^2 \hat{i} + t^3 \hat{j}) $

Computing the dot product gives:

$ P = (2t \cdot t^2) + (3t^2 \cdot t^3) = 2t^3 + 3t^5 $

The power generated at time $t$ is $P = (2t^3 + 3t^5) \, \text{W}$.