Two cyclists leave an intersection at the same moment. Cyclist A rides due east with a time-dependent speed u(t) m s−1, while Cyclist B rides due north with a time-dependent speed v(t) m s−1. Both speed functions are differentiable for t≥0, and the cyclists begin at the intersection.
(a) Derive, in terms of u(t), v(t), and t, a general formula for the rate at which the straight-line distance D(t) between the cyclists is changing. Present your answer in its simplest form without numerical substitution.
(b) Suppose u(t)=4+0.5t and v(t)=3. Describe qualitatively (no numerical calculation required) how the rate you obtained in part (a) behaves for t≥0, and explain whether the cyclists are moving farther apart or closer together as time progresses.
(c) Using correct units, explain how the sign of dtdD conveys information about whether the cyclists are approaching each other or separating.