Deriving Equations for Particle Motion and Momentum Conservation

[Ted123]

Homework Statement

[PLAIN]http://img51.imageshack.us/img51/4562/mecj.jpg

Homework Equations

The Attempt at a Solution

How do I go about showing LHS = RHS in each of these? (\wedge denotes cross product)

What is \dot{r}=\dot{x_1}-\dot{x_2} and \dot{R} ?

 

[HallsofIvy]

I would interpret the dot over the variable as being the derivative with respect to time (that's more a physics notation than mathematics). Thus \dot{r}=\dot{x_1}-\dot{x_2} is the rate at which the distance between the two objects is changing and \dot{R} is the speed at which their center of mass is moving.

 

[Ted123]

I would interpret the dot over the variable as being the derivative with respect to time (that's more a physics notation than mathematics). Thus \dot{r}=\dot{x_1}-\dot{x_2} is the rate at which the distance between the two objects is changing and \dot{R} is the speed at which their center of mass is moving.

So how do I find them?

ie. how do I differentiate R and r?

 

[Ted123]

Am I right in saying the LHS of (i) is

\displaystyle\frac{1}{2} (m_1 +m_2) \left| \frac{m_1\underline{\dot{x_1}}+m_2\underline{\dot{x_2}}}{m_1 + m_2} \right| ^2 + \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} \left| \underline{\dot{x_1}} - \underline{\dot{x_2}} \right| ^2 ?

If so, where do I go now?