Question

The trajectories of two particles moving in R3 are described by 10) = (a sin(e, sin(), 5 coses) and r2(t) = (sin(2t), 2 sin?(t), 2 cos(t)) for tER. a) Show that one of these trajectories lies on a sphere S centered at the origin in R3, and that the other one is contained in a plane. In what follows, we denote by r(t) the position of the particle that lies in a sphere. b) Prove that r(t) is orthogonal to the velocity v(t). c) Prove that the acceleration a(t) is directed towards the inside of the sphere S. Hint: Use the fact that r(t) is a vector normal to S and the dot product. d) Determine the speed of r(t) (that is the magnitude of v(t)). e) Deduce from the previous question that the distance d travelled by r(t) in one second (i.e. from to to to +1, for any to) satisfies the following series of inequalities: 2 < d < 2V2.