A particle moves in the plane along the path given by the vector function
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A particle moves in the plane along the path given by the vector function r(t)=⟨Rcost,Rsint⟩, where R>0 is a constant. The parameter t is measured in seconds.
(a) Determine whether the particle’s speed is constant. Support your conclusion with appropriate reasoning based on r(t).
(b) Describe how the direction of the unit tangent vector changes as the particle moves and explain what this indicates about the particle’s turning behavior along the path.
(c) Using your analysis from parts (a) and (b), find the radius of curvature at any point on the circle in terms of R and explain why this value is the same at every point on the path.