### Work Power Energy Question 2

#### Question 2 - 24 January - Shift 2

A body of mass $1 kg$ begins to move under the action of a time dependent force $\vec{F}=(t \hat{i}+3 t^{2} \hat{j}) N$. 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.

## Show Answer

#### Answer: (100)

#### Solution:

#### Formula: Work Done By A Variable Force

$ \vec{F}=t \hat{i}+3 t^{2} \hat{j} $

$ \frac{md \overrightarrow{{}v}}{dt}=t \hat{i}+3 t^{2} \hat{j} $

$m=1 kg, \int_0^{v} dv=\int_0^{t} tdt \hat{i}+\int_0^{t} 3 t^{2} dt \hat{j}$

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

Power $=\overrightarrow{{}F} \cdot \overrightarrow{{}V}=\frac{t^{3}}{2}+3 t^{5}$

At $t=2$, power $=\frac{8}{2}+3 \times 32$

$=100$