A particle moves according to the polar description r=2sin(3θ) with an angular position given by θ(t)=ln(1+2t) for t>0.
(a) Compute the velocity vector at t=0.5, expressing your answer in Cartesian coordinates.
(b) Compute the acceleration vector at t=0.5, again in Cartesian coordinates.
(c) Resolve the acceleration vector found in part (b) into components that are tangent and normal to the particle’s path at t=0.5.
(d) State whether the acceleration is perpendicular to the velocity at t=0.5, and provide a mathematical justification for your answer.