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## Homework Statement

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.

## Homework 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'

## 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.