Question

Consider a body of mass $2\text{kg}$ at rest at the origin at time $t=0$. A force $(\alpha t\hat{i}+\beta \hat{j})$ is applied on the body where $\alpha =2{\text{Ns}}^{-1}$ and $\beta =2\text{N}$. Then, which of the following statement(s) is/are true at $t=2\text{s}$?

• A. The velocity of the body along the direction $\hat{i}$ is $2\text{m/s}$.
• B. The velocity of the body along the direction $\hat{j}$ is $2\text{m/s}$.
• C. The velocity of the body along the direction $\hat{i}$ is $4\text{m/s}$.
• D. The velocity of the body along the direction $\hat{j}$ is $4\text{m/s}$.

Answer 1

Answer: (A), (B)

The velocity of the body along the direction $\hat{j}$ is $2\text{m/s}$.

Given that
Mass of the block $\left(m\right)=2\text{kg}$
Force $\left(F\right)=(\alpha t\hat{i}+\beta \hat{j})$
where
$\alpha =2{\text{Ns}}^{-1}$
$\beta =2\text{N}$
$F=\left(\alpha t\right)\hat{i}+\beta \hat{j}$
$\Rightarrow F=2t\hat{i}+2\hat{j}$
$\Rightarrow m\frac{dv}{dt}=2t\hat{i}+2\hat{j}$
On integrating,
$\Rightarrow {\int }_0^{v}mdv={\int }_0^t(2t\hat{i}+2\hat{j})dt$
[given, body is at rest at $t=0$]
$\Rightarrow mv=t^2\hat{i}+2t\hat{j}$
$\Rightarrow v=\frac{t^2\hat{i}+2t\hat{j}}{m}$
$\Rightarrow v=\frac{t^2\hat{i}+2t\hat{j}}{2}$
$\Rightarrow v=\frac{t^2}{2}\hat{i}+t\hat{j}$
$\Rightarrow v=2\hat{i}+2\hat{j}(\text{at}t=2\sec )$
$\therefore$ options (a) and (b) are correct.