Question

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

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

Answer 1

Answer: (D)

Applying Newton's second law of motion, acceleration, $\overrightarrow{a}=\frac{\overrightarrow{F}}{m}=2t\hat{i}+3t^2\hat{j}$

Acceleration is defined as rate of change of velocity,

$\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}$

$\overrightarrow{v}={\int }_0^t\overrightarrow{a}dt$

$\overrightarrow{v}={\int }_0^t(2t\hat{i}+3t^2\hat{j})dt$

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

Power, $P=\overrightarrow{F}.\overrightarrow{v}=(2t\hat{i}+3t^2\hat{j}).(t^2\hat{i}+t^3\hat{j})$

$P=(2t^3+3t^5)W$