At any instant the velocity of a particle of mass 500 g is $$(2t\hat{i} + 3t^2\hat{j})$$ m s$$^{-1}$$. 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:

Given: mass $$m = 500$$ g = 0.5 kg, velocity $$\vec{v} = (2t\hat{i} + 3t^2\hat{j})$$ m/s.

Acceleration:

$$ \vec{a} = \frac{d\vec{v}}{dt} = (2\hat{i} + 6t\hat{j})\;\text{m/s}^2 $$

At $$t = 1$$ s: $$\vec{a} = (2\hat{i} + 6\hat{j})$$ m/s².

Force:

$$ \vec{F} = m\vec{a} = 0.5(2\hat{i} + 6\hat{j}) = (\hat{i} + 3\hat{j})\;\text{N} $$

Comparing with $$(\hat{i} + x\hat{j})$$: $$x = 3$$.

The correct answer is $$x = 3$$.