#### Question: At any instant the velocity of a particle of mass $500 \mathrm{~g}$ is $\left(2 t \hat{i}+3 t^{2 \hat{j}}\right) \mathrm{ms}-1$. If the force acting on the particle at $\mathrm{t}=1 \mathrm{~s}$ is ${ }^{(\hat{i}+x \hat{j})} \mathrm{N}$. Then the value of $\mathrm{x}$ will be:

A) 2

B) 4

C) 6

D) 3

#### Answer: 3

#### Solution:

Given the velocity vector of a particle $v=\left(2 t \hat{i}+3 t^2 \hat{j}\right) \mathrm{ms}^{-1}$, the acceleration $a$ is the derivative of the velocity vector with respect to time. So, we have:

$$ a=\frac{d v}{d t}=(2 \hat{i}+6 t \hat{j}) \mathrm{ms}^{-2} $$

At $t=1 \mathrm{~s}$, the acceleration $a$ is $(2 \hat{i}+6 \hat{j}) \mathrm{ms}^{-2}$.

According to Newton’s second law, the force $F$ is equal to the mass $m$ times acceleration $a$. The mass $m$ is given as $500 \mathrm{~g}$, or equivalently, $0.5 \mathrm{~kg}$.

Therefore, the force $F$ on the particle at $t=1 \mathrm{~s}$ is:

$$ F=m \cdot a=0.5 \cdot(2 \hat{i}+6 \hat{j})=(1 \hat{i}+3 \hat{j}) \mathrm{N} $$

So, the force acting on the particle at $t=1 \mathrm{~s}$ is $(\hat{i}+x \hat{j}) \mathrm{N}$, where $x=3$.

Therefore, the answer is $x=3$.