Question 77: Laws of Motion

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 $$t=1 \mathrm{~s}$$ is $$(\hat{i}+x \hat{j}) \mathrm{N}$$. Then the value of $$x$$ will be:

Accepted Answer

Given the velocity vector of a particle $v = (2t \hat{i}+3 t^{2} \hat{j}) \, 	ext{ms}^{-1}$, the acceleration $a$ is the derivative of the velocity vector with respect to time. So, we have:
$a = \frac{dv}{dt} = (2 \hat{i} + 6t \hat{j}) \, 	ext{ms}^{-2}$.
At $t=1 \, 	ext{s}$, the acceleration $a$ is $(2 \hat{i} + 6 \hat{j}) \, 	ext{ms}^{-2}$.
According to Newton's second law, the force $F$ is equal to the mass $m$ times acceleration $a$. The mass $m$ is given as $500 \, 	ext{g}$, or equivalently, $0.5 \, 	ext{kg}$.
Therefore, the force $F$ on the particle at $t=1 \, 	ext{s}$ is:
$F = m \cdot a = 0.5 \cdot (2 \hat{i} + 6 \hat{j}) = (1 \hat{i} + 3 \hat{j}) \, 	ext{N}$.
So, the force acting on the particle at $t=1 \, 	ext{s}$ is $(\hat{i} + x \hat{j}) \, 	ext{N}$, where $x=3$.
Therefore, the answer is $x=3$.

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Laws of Motion

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

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