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(A) Given mass $m = 500 \, g = 0.5 \, kg$.The velocity vector is $\vec{v} = 2t \hat{i} + 3t^2 \hat{j}$.The acceleration $\vec{a}$ is the derivative of

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$3$

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(A) Given mass $m = 500 \, g = 0.5 \, kg$.The velocity vector is $\vec{v} = 2t \hat{i} + 3t^2 \hat{j}$.The acceleration $\vec{a}$ is the derivative of velocity with respect to time: $\vec{a} = \frac{d\vec{v}}{dt} = \frac{d}{dt}(2t \hat{i} + 3t^2 \hat{j}) = 2 \hat{i} + 6t \hat{j}$.At $t = 1 \, s$,the acceleration is $\vec{a} = 2 \hat{i} + 6(1) \hat{j} = 2 \hat{i} + 6 \hat{j} \, ms^{-2}$.Using Newton's second law,$\vec{F} = m\vec{a} = 0.5(2 \hat{i} + 6 \hat{j}) = 1 \hat{i} + 3 \hat{j} \, N$.Comparing this with the given force $(\hat{i} + x \hat{j}) \, N$,we get $x = 3$.

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

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