Meeting Times and Separation

Choose one positive direction, one origin and one clock for both particles. First write their positions s_P(t) and s_Q(t), including the starting offsets. They meet when s_P(t)=s_Q(t); equal velocities alone do not mean equal positions. The signed separation is d(t)=s_Q(t)-s_P(t), with d (t)=v_Q(t)-v_P(t). Physical separation is |d(t)|. To find its greatest value on a stated interval, check endpoints, equal-velocity times and any joins; interpret the result before or after a meeting as requested. For illustration, s_P=2t and s_Q=6-t describe a meeting at t=2, at position 4, although the velocities are never equal. The starting separation is the constant 6, not an integration constant to discard. If one particle starts later, its elapsed travelling time differs from the common clock. State the interval on which each expression applies and reject meeting times outside that interval. Numerical root finding must search the whole physical interval; retain guard digits until the final position or distance is calculated.