A 4 kg mass moves under the influence of a force $$\overrightarrow{F}=\left(4t^{3}\widehat{i}-3t\widehat{j}\right)N$$ where t is the time in second. If mass starts from origin at t= 0, the velocity and position after t= 2 s will be:

A mass $$m = 4$$ kg moves under force $$\vec{F} = (4t^3\hat{i} - 3t\hat{j})$$ N, starting from rest at the origin at $$t = 0$$. The acceleration is given by $$\vec{a} = \frac{\vec{F}}{m} = t^3\hat{i} - \frac{3t}{4}\hat{j}$$.

Integrating the acceleration vector from 0 to t yields the velocity $$\vec{v}(t) = \int_0^t \vec{a}\,dt' = \frac{t^4}{4}\hat{i} - \frac{3t^2}{8}\hat{j}$$, and at $$t = 2$$ one finds $$\vec{v} = 4\hat{i} - \frac{3}{2}\hat{j}$$.

Further integration from 0 to t yields the position $$\vec{r}(t) = \int_0^t \vec{v}\,dt' = \frac{t^5}{20}\hat{i} - \frac{t^3}{8}\hat{j}$$, which at $$t = 2$$ becomes $$\vec{r} = \frac{8}{5}\hat{i} - \hat{j}$$.

The position matches Option A: $$\vec{r} = \frac{8}{5}\hat{i} - \hat{j}$$. The correct answer is Option A.