Question

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

• A. $(2t^3+3t^4)\text{W}$
• B. $(2t^3+3t^5)\text{W}$
• C. $(2t^3+3t^3)\text{W}$
• D. $(2t^2+4t^4)\text{W}$

Answer 1

The correct option is B $(2t^3+3t^5)\text{W}$

since $\overrightarrow{F}=(2t\hat{i}+3t^2\hat{j})\text{N}$ and $m=1\text{kg}$
$\Rightarrow \overrightarrow{a}=\frac{\overrightarrow{F}}{m}=\frac{2t\hat{i}+3t^2\hat{j}}{1}=2t\hat{i}+3t^2\hat{j}{\text{m/s}}^2$
Integrating acceleration w.r.t time to find velocity at any instant $t$:
$\therefore \overrightarrow{v}={\int }_0^t\overrightarrow{a}dt$
$\overrightarrow{v}={\int }_0^{t}2tdt\hat{i}+{\int }_0^{t}3t^{2}dt\hat{j}$
$\therefore \overrightarrow{v}=t^2\hat{i}+t^3\hat{j}\text{m/s}$
From the definition of power $\left(P\right)$ delivered by the force:
$P=\overrightarrow{F}\cdot \overrightarrow{v}$
$\Rightarrow P=(2t\hat{i}+3t^2\hat{j})\cdot (t^2\hat{i}+t^3\hat{j})$
$\therefore P=2t^3+3t^5\text{W}$