Physics - W.E.P. Question with Solution | TestHub

Given mass $m=1$ kg and force $\vec{F}=t\hat{i}+2t^2\hat{j}$.

Acceleration of the body is given by Newton's second law:

$\vec{a}=\frac{\vec{F}}{m}=t\hat{i}+2t^2\hat{j}$

Since the body begins to move from rest, its velocity $\vec{v}$ at time $t$ is:

$\vec{v}=\int \vec{a}dt=\int (t\hat{i}+2t^2\hat{j})dt=\frac{t^2}{2}\hat{i}+\frac{2t^3}{3}\hat{j}$

Power developed by the force is the dot product of force and velocity:

$P=\vec{F}\cdot \vec{v}=(t\hat{i}+2t^2\hat{j})\cdot \left(\frac{t^2}{2}\hat{i}+\frac{2t^3}{3}\hat{j}\right)$

$P=\frac{t^3}{2}+\frac{4t^5}{3}$

Substituting $t=3$ s:

$P=\frac{3^3}{2}+\frac{4(3^5)}{3}=\frac{27}{2}+\frac{4\times 243}{3}$

$P=13.5+4\times 81=13.5+324=337.5\mathrm{W}$

Answer: $337.5\mathrm{W}$