# Proving a particle is moving in an elliptical orbit.

1. Apr 3, 2012

### GroupActiion

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

A particle of mass m is acted upon by two forces. P(t) in the x direction with magnitude p(sinkt) and Q(t) acting on the line y=x with magnitude qsinkt. At t=0 it starts at (b,0,0) and velocity p/(mk) moving toward the origin. Prove the particle is in an elliptical orbit.

2. The attempt at a solution
I have tried doing the problem in cartesian coordinates. The Q(t) force acts on the y=x line which makes a 45° angle with the x axis. Thus the force has a y and an x component. The y component is Q(t)sin°45 and the x component is Q(t)cos 45°. Adding up components in the y and x directions give two uncoupled differential equations(newton's second law). Problem is when I solve for the x(t) and y(t) along with the integration constants accounted for I get long expressions which do not make it obvious the particle is in an elliptical orbit i.e (x(t)2/b2 + y(t)2/b2 = 1). Am I on the right track? I do not need detailed descriptions just wondering if my analysis of the forces is right. Is there a way of analyzing the forces in polar coordinates because that way Q(t) acts in the radial direction and perhaps one would need to find the component of P(t) working the θ direction?

2. Apr 3, 2012

### tiny-tim

Welcome to PF!

Hi GroupActiion! Welcome to PF!

Hint: change the coordinates!