Question

A body of mass 1 kg begins to move under the action of a time dependent force $\vec{F} = ( 2\hat{i} + 3t^2 \hat{j} )N$, 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. $(2t^2 + 4t^4)W$
• B. $(2t^3 + 3t^4)W$
• C. $(2t^3 + 3t^5)W$
• D. $(2t^2 + 3t^3)W$

Answer 1

Answer: (C)

Applying Newton's second law of motion, acceleration, $\overrightarrow{ a }=\frac{\overrightarrow{ F }}{ m }=2 f +3 t ^{2 j }$
Acceleration is defined as rate of change of velocity,
$
\begin{array}{l}
\vec{a}=\frac{d \vec{v}}{d t} \\
\vec{v}=\int_{0}^{t} \vec{a} d t \\
\vec{v}=\int_{0}^{t}\left(2 \hat{i}^{2}+3 t^{2} \hat{j}\right) d t \\
\vec{v}=t^{2} \hat{i}+t^{3 \hat{j}}
\end{array}
$
Power, $P=\vec{F} \cdot \vec{v}=\left(2 f \hat{i}+3 t^{2} \hat{j}\right) \cdot\left(t^{2} \hat{i}+t^{3 \hat{j}}\right)$
$
P =\left(2 t ^{3}+3 t ^{5}\right) W
$