Question

A body of mass $1 \,kg$ begins to move under the action of a time dependent force $\vec{F}=\left(\hat{i}+3 t^2 \hat{j}\right) N$, where $\hat{i}$ and $\hat{j}$ are the unit vectors along $x$ and $y$ axis. The power developed by above force, at the time $t=2 s\,$, will be _________W.

Answer 1

$ \vec{ F }=t \hat{ i }+3 t ^2 \hat{ j } $
$ \frac{ md \vec{ v }}{ dt }=t \hat{ i }+3 t ^2 \hat{ j } $
$ m =1 \,kg , \int\limits_0^{\vec{v}} dv =\int\limits_0^{ t } tdt \hat{ i }+\int\limits_0^{ t } 3 t ^2 dt \hat{ j } $
$ \vec{ v }=\frac{ t ^2}{2} \hat{ i }+ t ^3 \hat{ j } $
Power $=\vec{ F } \cdot \vec{ V }=\frac{ t ^3}{2}+3 t ^5$
At $ t =2$, power $=\frac{8}{2}+3 \times 32$
$ =100$