Solution:

Given that the velocity \(v = \alpha \sqrt{x}\), we need to find the displacement \(x\) as a function of time \(t\).

1. \( v = \frac{dx}{dt} = \alpha \sqrt{x} \)

2. Rearranging and separating variables:
\[
\frac{dx}{\sqrt{x}} = \alpha \, dt
\]

3. Integrate both sides:
\[
2\sqrt{x} = \alpha t
\]

4. Solving for (x):
\[ x = \frac{\alpha^2 t^2}{4} \]

Thus, the displacement of the particle varies with time as \(x = \frac{\alpha^2 t^2}{4}\).