1. Given data

• Mass of body: $m=1\,\text{kg}$
• Force: $\vec F(t)= t\,\hat i + 3t^2\,\hat j$

We need the instantaneous power at $t=2\,\text{s}$.

2. Find acceleration

Using Newton's second law, $\vec F = m\vec a$ Since $m=1$ kg, $\vec a = \vec F = t\,\hat i + 3t^2\,\hat j$

So, $a_x=t,\qquad a_y=3t^2$

3. Find velocity as a function of time

Assuming the body begins to move from rest at $t=0$, $v_x=\int a_x\,dt = \int t\,dt = \frac{t^2}{2}$ $v_y=\int a_y\,dt = \int 3t^2\,dt = t^3$

Thus, $\vec v(t)=\frac{t^2}{2}\,\hat i + t^3\,\hat j$

4. Compute force and velocity at $t=2$ s

Force at $t=2$: $\vec F(2)=2\,\hat i + 3(2)^2\,\hat j=2\,\hat i+12\,\hat j$

Velocity at $t=2$: $\vec v(2)=\frac{(2)^2}{2}\,\hat i + (2)^3\,\hat j=2\,\hat i+8\,\hat j$

5. Instantaneous power

Power is given by $P = \vec F \cdot \vec v$ So, $P = (2\,\hat i+12\,\hat j)\cdot(2\,\hat i+8\,\hat j)$ $P = 2\cdot 2 + 12\cdot 8$ $P = 4 + 96 = 100\,\text{W}$

6. Final answer

$\boxed{100}$

The derived answer matches the stored correct answer.