The velocity of a particle is $v=v_0+gt+f{t}^2$. If its position is x=0 at t=0, then its displacement after unit time $\left(t=1\right)$ is:

Given, velocity $v=v_0+gt+f{t}^2\therefore v=\frac{dx}{dt}\text{ or }{\int }_0^x dx={\int }_0^t vdt$

Or $x={\int }_0^t \left(v_0+gt+f{t}^2\right)dtx=v_{0}t+\frac{g{t}^2}{2}+\frac{f{t}^3}{3}+C$ the constant of integration.

Given, $x=0,t=0\therefore C=0\text{ Or }x=v_{0}t+\frac{g{t}^2}{2}+\frac{f{t}^3}{3}\text{ At }t=1\sec \therefore x=v_0+\frac{g}{2}+\frac{f}{3}$