# Separate the variables by using kinetic energy and potential energy

Is it possible to solve $$x^2\ddot{x}=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}m}$$ to get x(t)? I can't see how!! Maybe I'm just missing something...

## Answers and Replies

Tomsk said:
Is it possible to solve $$x^2\ddot{x}=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}m}$$ to get x(t)? I can't see how!! Maybe I'm just missing something...

You can separate the variables by using kinetic energy and potential energy, and conservation of angular momentum. I did that. I put
it all into polar coordinates - and I came with an integral of a function
of r (radius) that there's probably a formula for somewhere. It's not
an easy integral though.

It's simpler if you assume the particles don't have any angular momentum relative to each other. That's how I got curious about it - I did an exercise about the enormous acceleration a proton would have, jetting out of the nucleus, if there weren't any strong nuclear forces holding it
in. So I wondered, what's its equation of motion?

You could figure out the proton's final velocity without doing any
complicated integrals - that would be (sort of) interesting too.

Laura

arildno
Science Advisor
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Certainly.
Let the squared position stand in the denominator on the right-hand side, and multiply your diff. eq. with the velocity.
You now will get a first integral (take note of the sign of the square root used!), this can be integrated one more time.

Spooky....

Well, I got
$$x^3=-\frac{9}{2}\frac{q_{1}q_{2}}{4\pi\epsilon_{0}m}t^2$$

The sign threw me though, so I'm not sure of it.

Thanks! I might try it relativistically, to stop my brain from rotting before I go back to uni.

One thing, there's only one m taken into account, which must be the mass of the particle which moves a distance x, or is x the distance between the two particles? I'm assuming they're both free to move, so wouldn't you need to take both masses into account? Hmmm

in two dimension this is known as the one body problem (using a equilvalent equation in 2D)... you might wanna try to solve that (it is impossible to get r(t) explicitly, but you can find out the shape of the orbit) its a lot of "fun". after that go to two body problem... more fun awaits...