A body of mass $$2$$ kg is initially at rest. It starts moving unidirectionally under the influence of a source of constant power $$P$$. Its displacement in $$4$$ s is $$\frac{1}{3}\alpha^2\sqrt{P}$$ m. The value of $$\alpha$$ will be ______.

We have a body of mass $$m = 2$$ kg starting from rest under constant power $$P$$. We need to find the displacement in 4 seconds.

Under constant power, $$P = Fv = mav = mv\frac{dv}{dt}$$, so $$P \, dt = mv \, dv$$. Integrating both sides from $$t = 0$$ (where $$v = 0$$) to time $$t$$:

$$Pt = \frac{1}{2}mv^2$$

This gives $$v = \sqrt{\frac{2Pt}{m}} = \sqrt{\frac{2Pt}{2}} = \sqrt{Pt}$$.

Now since $$v = \frac{dx}{dt} = \sqrt{P} \cdot t^{1/2}$$, we integrate to find displacement:

$$x = \int_0^4 \sqrt{P} \cdot t^{1/2} \, dt = \sqrt{P} \cdot \left[\frac{2t^{3/2}}{3}\right]_0^4 = \sqrt{P} \cdot \frac{2 \times 8}{3} = \frac{16\sqrt{P}}{3}$$

We are told that displacement equals $$\frac{1}{3}\alpha^2\sqrt{P}$$. Equating:

$$\frac{1}{3}\alpha^2\sqrt{P} = \frac{16\sqrt{P}}{3}$$

$$\alpha^2 = 16$$

So, the answer is $$\alpha = 4$$.