How to Derive Equations of Motion for Colliding Masses in a Central Field?

[PhysStudent81]

Homework Statement

I have two masses of finite width, m_1 and m_2. The force is Newton's gravity, so U = k/r. I want to work out their relevant equations of motions r_1(t) and r_2(t) as they start off from rest and collide. I don't want to consider any rotational motion.

Homework Equations

U = \frac{k}{r}
r = r_{1}(t) - r_{2}(t)

0.5μ\dot{r}^2 = E_{tot} - \frac{k}{r}

dt = \frac{μ}{2}∫\frac{dr}{\sqrt{E_{tot} - \frac{k}{r}}}

The Attempt at a Solution

I try to integrate the above equation it gives me something very complicated (I end up integrating cosec^3 after making the substitution \frac{1}{r} = \sin^{2}(\theta)) which gives me t = t(r), but I can't invert this to give me r = r(t).

Am I doing something wrong? Is there another way of doing it that doesn't involve lagrangian or hamiltonian dynamics (which I haven't studied).

Another way would be to solve the the 2nd order differential equation directly:

\frac{dr^{2}}{dt^{2}} = \frac{k}{r^{2}}

but I can't seem to do this (I fee I'm missing something very simple here). I know that if I let r = At^{\frac{2}{3}} this is a solution but it doesn't have enough constants.

Any pointers?

Thanks,

Rob

 

[bp_psy]

Try this thread.

https://www.physicsforums.com/showthread.php?p=2497264#post2497264

 

[TSny]

I try to integrate the above equation it gives me something very complicated (I end up integrating cosec^3 after making the substitution \frac{1}{r} = \sin^{2}(\theta)) which gives me t = t(r), but I can't invert this to give me r = r(t).

Am I doing something wrong?

I think what you are doing is correct. Did you allow for the fact that both k and E_tot are negative numbers? Anyway, as you say, you can get an expression for t(r). I agree that the result looks too complicated to invert for r(t).

 

[PhysStudent81]

Try this thread.

https://www.physicsforums.com/showthread.php?p=2497264#post2497264

oh my goodness thank you.

Rob

 

[PhysStudent81]

So strange that for such seemingly simple problem there is no closed form result for r(t). t(r) is actually all I wanted so that's ok!