Step 1: Understanding the Concept:
Power developed by a force is the dot product of the force vector and the instantaneous velocity vector: $P = \vec{F} \cdot \vec{v}$. We must find velocity by integrating acceleration ($F/m$).
Step 2: Key Formula or Approach:
1. Newton's second law: $\vec{a} = \frac{\vec{F}}{m}$
2. Kinematic integration: $\vec{v} = \int \vec{a} \, dt$
3. Dot product power: $P = \vec{F} \cdot \vec{v}$
Step 3: Detailed Explanation:
Given $m = 1\text{ kg}$. So, $\vec{a} = \vec{F} = (t\hat{i} + 2t^2\hat{j})$.
1. Integrate to find velocity (assuming starting from rest):
\[ \vec{v} = \int (t\hat{i} + 2t^2\hat{j}) \, dt = \frac{t^2}{2}\hat{i} + \frac{2t^3}{3}\hat{j} \]
2. Calculate instantaneous power:
\[ P = \vec{F} \cdot \vec{v} = (t\hat{i} + 2t^2\hat{j}) \cdot \left( \frac{t^2}{2}\hat{i} + \frac{2t^3}{3}\hat{j} \right) \]
\[ P = \frac{t^3}{2} + \frac{4t^5}{3} \]
3. Substitute $t = 3\text{ s}$:
\[ P(3) = \frac{3^3}{2} + \frac{4 \times 3^5}{3} = \frac{27}{2} + 4 \times 81 = 13.5 + 324 = 337.5\text{ W} \]
Step 4: Final Answer:
The power developed at $t = 3\text{ s}$ is $337.5\text{ W}$.