Two cyclists start from the same point on a straight
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
Two cyclists start from the same point on a straight, level road at time t=0 seconds and ride along the positive x–axis. Cyclist A’s velocity (in meters per second) is modeled by vA(t)=3+0.4t, while Cyclist B’s velocity is modeled by vB(t)=5−0.2t for 0≤t≤20. Assume both velocity models remain valid and that neither cyclist turns around.
(a) Describe, in words, what it would mean for there to exist a time t=c with 0<c<20 such that vA(c)=vB(c). Does equality of the velocities alone guarantee that the cyclists are side by side? Explain your reasoning without performing any calculations.
(b) Explain why the difference function vB(t)−vA(t) represents the instantaneous rate of change of the separation distance between the cyclists. Include proper units in your explanation.
(c) Suppose it is known (without calculation) that vB(t)−vA(t) is negative for some interval 0<t<T and positive for some t>T. Interpret this fact in terms of the cyclists’ motion and their relative positions on the road.