The shortest time

Position runs across and speed runs up; the origin means at rest on the stop line. Drag the starting point: the gold path is the quickest one. The force is always at full strength, forward or back, and reverses where the path meets the gold switching curve, which then leads home. On the right, the ice-blue λ₂, the costate, crosses zero just when the force reverses. The ice-blue contours join points with the same time to go, half a unit apart: drag the start outwards across one and the time grows by 0.5. The bottom-right corner measures the time on the drawn path (with a force of size 1 the speed changes by 1 each second, so the total rise and fall of the speed is the time) and works it out again from Bushaw's formula. The closer the start is to the gold curve, the sooner the switch; right on it, there is none at all. Switch to the block on a spring: the path becomes a chain of circular arcs, the switches line up on a chain of semicircles, and between two switches exactly half a period, π, goes by. Drag the start about and count the switches.