# Central Force Motion (two-mass orbit problem)

1. Apr 10, 2008

### mat5041

1. The problem statement, all variables and given/known data
1a) Verify that the kinetic energy of two particles can be written as the kinetic energy of a single 'reduced mass' particle with velocity r'equal to their difference.

1b) In the center-o-mass frame, show that theta is the same for r1 as for r, whereas the polar angle for r2 differs by pi radians from that of r

1c)Show that if r(theta) is an ellipse, then r1(theta) is an ellipse with the center of mass at one focus; r2(theta) is an ellipse with one focus at the center of mass; in a (non-inertial) frame in which the Earth is at rest, the sun would move in an ellipse around the sun if they were the only two massive objects.

2. Relevant equations
1a)
kinetic energy of the system:
T=1/2*m1*|r'1|^2+1/2*m2*|r'2|^2

reduced mass (greek letter mu=u)
u=m1*m2/(m1+m2)

relation of r, r1, and r2
r=r1-r2

1b)
I think this involves drawing a vector diagram of r, r1, and r2 and figuring out the angles, I'm not sure which angles to consider though or how to show that they differ by pi so I think the formulas involve law of sines, law of cosines, sin, and cos...

1c) eq of ellipse:
alpha/r=1+epsilon*cos(theta)
alpha=l^2/(u*k) k is a constant
epsilon= sqrt(1+(2*E*L^2/(u*k^2))
L=u*r^2*theta'

3. The attempt at a solution
1a) I got this, It just involved a bunch of algebra and substitutions. I can elaborate if needed.

1b) As I said in the equations part, I believe this involves the vector diagram of the two-mass system but I'm unsure how to prove the difference of the polar angles is equal to pi

1c) I'm not entirely sure where to start on this, I began substituting everything into the equation of an ellipse but I didn't get anywhere with that.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution