Define acceleration.

Acceleration ($a$) is the rate of change of velocity. If $u$ is the initial velocity and $v$ is the final velocity over a time $t$, then:

$$a = \frac{v - u}{t}$$

Rearranging this equation to solve for the final velocity $v$:

$$v = u + at$$

For motion with constant acceleration, the average velocity ($\bar{v}$) is the arithmetic mean of the initial and final velocities:

$$\bar{v} = \frac{u + v}{2}$$

$$\bar{v} = \frac{u + (u + at)}{2}$$ $$\bar{v} = \frac{2u + at}{2}$$ $$\bar{v} = u + \frac{1}{2}at$$

Displacement ($S$) is the product of average velocity and time:

$$S = \bar{v}t$$

$$S = \left(u + \frac{1}{2}at\right)t$$ $$S = ut + \frac{1}{2}at^2$$

The question uses $V$ for initial velocity instead of $u$. Replacing $u$ with $V$:

$$S = Vt + \frac{1}{2}at^2$$

This is the derived equation.