Minimum and maximum values

Question

A particle moves around the ellipse (y/3)² + (z/2)² = 1

in the yz-plane in such a way that its position at a time t is r(t) = (3 cos t)j + (2 sin t)k.

Find the maximum and minimum values of |v| and |a|

Answer 1

For r(t) = 3 cost j +2 sint k v(t)=-3 sint j +2 cost k and a(t) = -3 cost j -2 sint k. |v|² = 9 sin²t + 4 cos²t and |a|²= 9 cos²t + 4 sin²t. Now d|v|²/dt =18 sint cost -8 sint cost = 10 sint cost and d|a|²/dt = 18 sint cost -8 cost sint = 10 sint cost. Using these 10 sint cost =0 or 5sin2t=0 and t= 0,π, 3π/2. So looking at |v|² for t=0,π v²(0) and v²(π) = 4 and |v| =2 and v²(3π/2) = 9 and |v|=3. So the min is 2 and the max is 3. Doing the same for |a|² gives a min of 2 and a max of 3 for |a|.