Modelling an interation between two protons

[kaspis245]

Homework Statement

A proton with velocity $v$ is fired from infinity in the direction of another free and stationary proton as shown in the picture. Find the minimum distance between the protons.

Relevant Equations

Coulomb's law: $F=\frac{1}{4πε_ο}\frac{q^2}{r^2}$

The image given with the problem:

I found the original problem too difficult to solve, so I thought I'd make it easier by first considering the stationary proton as fixed in place. However, this too proved to be way more difficult than I expected. Here's the updated diagram I used in my attempt to solve it:

I started by writing the force felt by the moving proton at all times:
$$\mathbf F = \frac{1}{4πε_ο}\frac{q^2}{r^2} \hat {\mathbf r}=\frac{1}{4πε_ο}\frac{q^2}{r^3}(L\hat {\mathbf x}+R\hat {\mathbf y})$$

From this point forward I'm struggling to find any conservation law or point of equilibrium that I could use. I know that conservation of momentum yields
$$\Delta p = -mv \hat {\mathbf x} + \int \mathbf F \, dt $$

unfortunately, I don't see how's that possible to solve or how does that even help me to arrive at the solution.

Does anybody have any ideas?

 

[PeroK]

Homework Statement:: A proton with velocity $v$ is fired from infinity in the direction of another free and stationary proton as shown in the picture. Find the minimum distance between the protons.
Relevant Equations:: Coulomb's law: $F=\frac{1}{4πε_ο}\frac{q^2}{r^2}$

The image given with the problem:
View attachment 265816
I found the original problem too difficult to solve, so I thought I'd make it easier by first considering the stationary proton as fixed in place. However, this too proved to be way more difficult than I expected. Here's the updated diagram I used in my attempt to solve it:
View attachment 265819

I started by writing the force felt by the moving proton at all times:
$$\mathbf F = \frac{1}{4πε_ο}\frac{q^2}{r^2} \hat {\mathbf r}=\frac{1}{4πε_ο}\frac{q^2}{r^3}(L\hat {\mathbf x}+R\hat {\mathbf y})$$

From this point forward I'm struggling to find any conservation law or point of equilibrium that I could use. I know that conservation of momentum yields
$$\Delta p = -mv \hat {\mathbf x} + \int \mathbf F \, dt $$

unfortunately, I don't see how's that possible to solve or how does that even help me to arrive at the solution.

Does anybody have any ideas?

If you assume one charge is fixed, you can use conservation of energy and angular momentum.

If you assume both charges can move, then consider the collision in the centre of momentum frame of reference.

 

[haruspex]

Homework Statement:: A proton with velocity $v$ is fired from infinity in the direction of another free and stationary proton as shown in the picture.

any ideas?

Consider what the trajectories are for two bodies attracting by an inverse square law. What does that suggest if they repel?

 

[kaspis245]

Consider what the trajectories are for two bodies attracting by an inverse square law. What does that suggest if they repel?

Intuition suggests that it's a parabolic path, but I don't know how to express it.

 

[PeroK]

Intuition suggests that it's a parabolic path, but I don't know how to express it.

The key, as always, is conservation of energy and angular momentum.

 

[haruspex]

Intuition suggests that it's a parabolic path, but I don't know how to express it.

Parabolas have one fewer degrees of freedom than ellipses. Think of another conic.

 

[kaspis245]

Parabolas have one fewer degrees of freedom than ellipses. Think of another conic.

Well then, I suppose it's a hyperbola. Is there a way I can formally prove this?

 

[haruspex]

Well then, I suppose it's a hyperbola. Is there a way I can formally prove this?

Of course, but it probably isn't that helpful in solving the given question. Just thought it might help you see what is going on.
Why not follow Perok's suggestions in post #2?