A motion-tracking camera is stationed north of a straight east–west racetrack
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A motion-tracking camera is stationed 20m north of a straight east–west racetrack. A sprinter runs eastward along the track at a speed that increases uniformly from 8ms−1 to 11ms−1 during a 6-second interval that begins when the runner is due west of the camera. Let θ(t) be the instantaneous bearing (angle measured clockwise from due north) from the camera to the runner.
(a) Describe qualitatively how θ changes—including whether it increases or decreases—during the interval while the runner passes the camera’s line of sight. Support your answer with geometric reasoning based on the configuration of the track and camera.
(b) Using appropriate variables and symbolic work, develop an expression for dtdθ in terms of the runner’s eastward position x (measured from the point closest to the camera) and velocity dtdx. Do not substitute numerical values yet.
(c) Evaluate your expression at the instant when x=0 (the runner is directly south of the camera) and interpret the sign and magnitude of dtdθ in the context of the camera’s panning motion.