A particle moves in space so that its position at time t seconds is given by the vector-valued function
r(t)=3mcos(t)sin(t)34t
where the horizontal components are measured in metres and t≥0.
(1) Part 1: (a) Calculate the particle’s velocity vector v(t) and acceleration vector a(t).
(2) Part 2: (b) Show that the particle’s speed is the same for all values of t, and determine its numerical value including units.
(3) Part 3: (c) Determine whether the velocity vector is perpendicular to the acceleration vector for all t. Justify your answer with an appropriate calculation.
(4) Part 4: (d) Describe the geometric shape of the path traced by r(t) and explain how your answers to parts (b) and (c) support your description.