A body of mass \(2\) kg begins to move under the influence of time dependent force \(\vec{F} = (2t\hat{i} + 6t^2\hat{j})\) N, where \(\hat{i}\) and \(\hat{j}\) are unit vectors along \(x\) and \(y\)-axis respectively. The power produced by the force at \(t = 2\) s is _____ W.

Given \(\vec{F} = (2t\hat{i} + 6t^2\hat{j})\) N and \(m = 2\) kg.

Acceleration of the body is \(\vec{a} = \dfrac{\vec{F}}{m} = \dfrac{2t\hat{i} + 6t^2\hat{j}}{2} = t\hat{i} + 3t^2\hat{j}\) m/s\(^2\).

Since the body begins to move from rest, its velocity at time \(t\) is given by \(\vec{v} = \int \vec{a} dt = \int (t\hat{i} + 3t^2\hat{j}) dt = \dfrac{t^2}{2}\hat{i} + t^3\hat{j}\) m/s.

At \(t = 2\) s, the force is \(\vec{F} = 2(2)\hat{i} + 6(2)^2\hat{j} = 4\hat{i} + 24\hat{j}\) N.

At \(t = 2\) s, the velocity is \(\vec{v} = \dfrac{(2)^2}{2}\hat{i} + (2)^3\hat{j} = 2\hat{i} + 8\hat{j}\) m/s.

Power produced by the force is \(P = \vec{F} \cdot \vec{v} = (4\hat{i} + 24\hat{j}) \cdot (2\hat{i} + 8\hat{j})\).

\(P = (4 \times 2) + (24 \times 8) = 8 + 192 = 200\) W.

Answer: \(200\)