# Two-dimensional motion under Central Force (1 Viewer)

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#### Eagletsam

1. The problem statement, all variables and given/known data

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).

2. Relevant 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?

3. 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.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

#### fzero

Homework Helper
Gold Member
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!

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