Some students have just drawn my attention to the problem of particle motion under central force where the x and y coordinates are specified as functions of time, such as
x(t) = A [ kt – cos(βt) ]
y(t) = B [ 1 – sin(βt) ],
(here A, B, β and k are constants).
The problem is to determine the radial and tangential orbital velocities (orbit assumed to be elliptical), and recover the canonical form of the equation (x2/a2 + y2/b2 = 1).
I have tried to figure this out for a few days without much success. Can anyone assist please?
The Attempt at a Solution
Express position vector r as
r = [x2 + y2]1/2
and evaluate dr/dt to obtain velocity as function of t. But this leads to messy calculation, which does not yield the angular and radial dependence.
Equally, attempt to eliminate t and obtain an equation in x and y proves quite hard.
We have also thought of coordinate transformation from (x,y,a,b) to (u,v,c,d) where u,v are velocity axes, but could not quite effect that.