Two-dimensional motion under Central Force

[Eagletsam]

Homework Statement

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).

Homework Equations

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.

 

[fzero]

Since x(t) is unbounded, this doesn't look like a closed orbit. It's possible that this is a parabolic solution to a central force law, but you might want to verify that.

 

[Eagletsam]

Thanks, very much, fzero, for early intervention.

I agree entirely that the orbit may not be elliptical!