Given the velocity function: $$v = \alpha \sqrt{x}$$

We know that velocity $v$ is the rate of change of displacement with respect to time:

$$\therefore \frac{dx}{dt} = \alpha \sqrt{x}$$

Rearranging and separating variables, we get:

$$\Rightarrow \frac{dx}{\sqrt{x}} = \alpha \, dt$$

To solve this, we integrate both sides:

$$\int\limits_{0}^{x} \frac{dx}{\sqrt{x}} = \alpha \int\limits_{0}^{t} dt$$

This gives us:

$$\Rightarrow \left[ \frac{2\sqrt{x}}{1} \right]_{0}^{x} = \alpha \left[t\right]_{0}^{t}$$

Simplifying, we obtain:

$$\Rightarrow 2\sqrt{x} = \alpha t $$

Squaring both sides:

$$\Rightarrow x = \frac{\alpha^2}{4} t^2$$

Thus, the displacement $x$ varies with time $t$ as:

$$x \propto t^2$$