The velocity of a particle is given by $ v = v_0 + gt + Ft^2 $. Its position is $ x = 0 $ at $ t = 0 $. To find its displacement after time $ t = 1 $, follow these steps:

Given:

$ v = v_0 + gt + Ft^2 $

We know that:

$ \frac{dx}{dt} = v_0 + gt + Ft^2 $

To find the displacement, integrate both sides with respect to $ t $:

$ \int_{x = 0}^{x} dx = \int_{t = 0}^{t = 1} (v_0 + gt + Ft^2) \, dt $

This simplifies to:

$ x = \left[ v_0 t + \frac{gt^2}{2} + \frac{Ft^3}{3} \right]_{t = 0}^{t = 1} $

Evaluating the integral from $ t = 0 $ to $ t = 1 $:

$ x = v_0 + \frac{g}{2} + \frac{F}{3} $

Therefore, the displacement after time $ t = 1 $ is:

$ x = v_0 + \frac{g}{2} + \frac{F}{3} $