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.

The instantaneous power delivered by a force is defined as the scalar (dot) product of the force and the velocity of the body at that instant:
$$P(t)=\vec F(t)\cdot\vec v(t)\quad -(1)$$

We are given the time-dependent force
$$\vec F(t)=\bigl(2t\,\hat i+6t^{2}\,\hat j\bigr)\,{\rm N}$$
and the mass of the body $$m=2\;{\rm kg}$$. To use equation $$(1)$$ we first need the velocity $$\vec v(t)$$.

Step 1: Find the acceleration.
Newton’s second law gives
$$\vec a(t)=\frac{\vec F(t)}{m}=\frac{2t}{2}\,\hat i+\frac{6t^{2}}{2}\,\hat j=t\,\hat i+3t^{2}\,\hat j\;{\rm m\,s^{-2}}.$$

Step 2: Integrate acceleration to get velocity.
The problem states “a body begins to move under the influence of the force,” so we take the initial velocity at $$t=0$$ to be zero: $$\vec v(0)=\vec 0$$.
Integrating component-wise, $$ v_x(t)=\int_{0}^{t}t'\,dt'=\frac{t^{2}}{2},\qquad v_y(t)=\int_{0}^{t}3{t'}^{2}\,dt'=t^{3}. $$ Hence $$\vec v(t)=\frac{t^{2}}{2}\,\hat i+t^{3}\,\hat j\;{\rm m\,s^{-1}}.$$

Step 3: Evaluate force and velocity at $$t=2\;{\rm s}$$.
Force:
$$\vec F(2)=\bigl(2\!\times\!2\,\hat i+6\!\times\!2^{2}\,\hat j\bigr) =(4\,\hat i+24\,\hat j)\;{\rm N}.$$ Velocity:
$$\vec v(2)=\frac{2^{2}}{2}\,\hat i+2^{3}\,\hat j =(2\,\hat i+8\,\hat j)\;{\rm m\,s^{-1}}.$$

Step 4: Compute the power using equation $$(1)$$.
$$ P(2)=\vec F(2)\cdot\vec v(2) =(4\,\hat i+24\,\hat j)\!\cdot\!(2\,\hat i+8\,\hat j) =4\times2+24\times8 =8+192 =200\;{\rm W}. $$

Therefore, the power produced by the force at $$t=2$$ s is 200 W.