Ambush chase kinematics. The predator accelerates at a up to top speed v_max: x_p(t) = a t^2 / 2 for t ≤ v_max/a, then linear. The prey starts d metres ahead, reacts after t_r seconds and accelerates at a_p to v_p toward a refuge D_r metres away. Capture is the first time the gap d + x_q - x_p reaches zero before the prey reaches the refuge and before the predator's sprint stamina S runs out.

Allosaurus speeds are not directly measured; the defaults (about 8-10 m/s top speed, a few m/s² acceleration) are rough, order-of-magnitude estimates from biomechanical scaling studies and could be quite wrong. Treat the numbers as a model exercise, not palaeontological fact.