A particle moves in space such that its distance from the origin is always
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A particle moves in space such that its distance from the origin is always 7 m. Let r(t) denote its position at time t seconds.
(a) Prove that the velocity vector v(t) is perpendicular to the position vector r(t) for every instant of time.
(b) Show that the acceleration vector a(t) can be decomposed into radial and tangential components of the form a(t)=α(t)r(t)+β(t)v(t), and determine explicit formulas for the scalar functions α(t) and β(t).
(c) Explain how the sign of α(t) affects whether the particle’s speed is increasing, decreasing, or remaining constant.