A particle moves in three–dimensional space with position vector r(t)= e^ t , ; t^ 2 , ; t
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A particle moves in three–dimensional space with position vector
r(t)=⟨et,t2,sint⟩,t≥0.
Answer each part thoroughly, supporting all conclusions algebraically.
(a) Obtain explicit formulas for the velocity vector v(t) and acceleration vector a(t).
(b) Derive a simplified expression for v(t)⋅a(t).
(c) Determine all times t>0 at which the particle’s speed is momentarily constant. Justify that your solution set is complete.
(d) Compute the acute angle between v(t) and a(t) at t=0, expressing the angle in radians.