Trigonometry · Superchallenge problem

Two straight roads cross at a junction $O$ at a $60^\circ$ angle. At noon, car $P$ is $10$ km from $O$, driving along its road toward $O$ (and then on through it) at $40$ km/h. At the same moment car $Q$ leaves $O$ along the other road at $20$ km/h, heading into the $60^\circ$ angle formed with $P$'s side of the junction. Both keep constant speeds. Find the smallest distance between the cars, in km, in simplest radical form.