A pursuit problem: Daniel and the Lion The luckless Daniel (D) is thrown into a circular arena of radius a containing a lion (L). Initially the lion is at the centre O of the arena while Daniel is at the perimeter. Daniel's strategy is to run with his maximum speed u around the perimeter. The lion responds by running at its maximum speed U in such a way that it remains on the (moving) radius O D.
Show that r, the distance of L from O, satisfies the differential equation
r·2 = (u2/a2)((U2a2/u2) - r2).
Find r as a function of t. If U ≥ u, show that Daniel will be caught, and find how long this will take. Show that the path taken by the lion is an arc of a circle. For the special case in which U = u, sketch the path taken by the lion and find the point of capture.