A constant force F = (4 i + 2 j ) N acts on a particle. The...

The position vector of the particle is given by $\vec{r} = (t^{3}\hat{i} - 2t^{2}\hat{j})\text{ m}$.

The velocity of the particle is the rate of change of its position vector:

$\vec{v} = \dfrac{d\vec{r}}{dt} = \dfrac{d}{dt}(t^{3}\hat{i} - 2t^{2}\hat{j}) = (3t^{2}\hat{i} - 4t\hat{j})\text{ m/s}$

At $t = 2\text{ s}$, the velocity is:

$\vec{v} = 3(2)^{2}\hat{i} - 4(2)\hat{j} = (12\hat{i} - 8\hat{j})\text{ m/s}$

The force acting on the particle is $\vec{F} = (4\hat{i} + 2\hat{j})\text{ N}$.

The instantaneous power delivered by the force is given by the dot product of force and velocity:

$P = \vec{F} \cdot \vec{v} = (4\hat{i} + 2\hat{j}) \cdot (12\hat{i} - 8\hat{j})$

$P = (4)(12) + (2)(-8) = 48 - 16 = 32\text{ W}$

Answer: $32\text{ W}$