$1.2.8^*.$ The particle, after leaving the source, flies at a constant speed for a distance $L$, and then decelerates with acceleration $a$. At what speed will the particle have the shortest travel time from its departure to its stop?
On a constant velocity interval, the velocity is:
$$v_0 = \frac{L}{t}$$
During the deceleration interval the velocity changes as
$v(t) = v_0-at \Rightarrow v_0=at$
Equating the equations, we obtain
$L=at^2$
From where
$v_0 = \sqrt{La}$
$$v = \sqrt{La}$$