Question

A force $\overrightarrow{F}=6t^2\hat{i}+4t\hat{j}$ is acting on a particle of mass $3kg$ then what will be velocity of particle at $t=3$ second and if at $t=0$, particle is at rest :

• A. $18\hat{i}+6\hat{j}$
• B. $18\hat{i}+12\hat{j}$
• C. $12\hat{i}+6\hat{j}$
• D. None of these

Answer 1

The correct option is A $18\hat{i}+6\hat{j}$

$\text{Step 1: Newton's second law for finding Acceleration}$

$\sum \overrightarrow{F}=m\overrightarrow{a}$

$\Rightarrow \overrightarrow{a}=\frac{F}{m}=\frac{1}{3}\overrightarrow{F}=2t^2\hat{i}+\frac{4}{3}t\hat{j}m/s^2$

$\text{Step 2: Velocity calculation using Acceleration}$

Acceleration is also given by

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

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

Integrating both sides

Limits for $v$ is $0$ to $v$ and for $t$ is $0$ to $3s$

$\therefore {\int }_0^{v}dv={\int }_0^3(2t^2\hat{i}+\frac{4}{3}t\hat{j})dt$

$\Rightarrow$ $\overrightarrow{v}={[\frac{2t^3}{3}\hat{i}+\frac{4}{3}\frac{t^2}{2}\hat{j}]}_0^3$

$\Rightarrow$ $\overrightarrow{v}=18\hat{i}+6\hat{j}m/s$

Hence option $A$ is correct.