Rail Car with a Sail in the Wind

[haruspex]

Please prove your statement, it is totally not obvious to me.

But I agree that fewer claims is better, so let's write equations without it:
1. Momentum conservation parallel to sail (no friction with sail)
$m_p\vec{v}_{p \perp s} = m_p\vec{v}_{p \perp s}^1$
2. Momentum conservation along rail
$m_p\vec{v}_{p \parallel r} = m_p\vec{v}_{p \parallel r}^1 + m_c\vec{v}_c$
3. Restricton
$<\vec{v}_c, \vec{r}_{\perp}> = 0$
4. Energy conservation
$m_p\left| \vec{v}_p \right|^2 = m_p\left|\vec{v}_p^1\right|^2 + m_c\left|\vec{v}_c\right|^2$

Do you agree with these 4 euations?

Yes, those are the correct equations.
In the square-on case, your equation 1 gives $m_p\vec{v}_{p \perp s} = m_p\vec{v}_{p \perp s}^1=0$. I.e., it bounces back along the same line.

 

[haruspex]

The inertial frame of reference associated with the car before the collision is not accelerate.

Yes, but previously you did not specify "before the collision". That's what has confused some commentators, me included. I realized that's what you probably meant a while back so stopped drawing attention to it.
Since the problem statement says the cart starts at rest, you are in fact using the ground frame, so why not say so?

 

[MaratZakirov]

Yes, but previously you did not specify "before the collision". That's what has confused some commentators, me included. I realized that's what you probably meant a while back so stopped drawing attention to it.
Since the problem statement says the cart starts at rest, you are in fact using the ground frame, so why not say so?

Let's look at my first post

All velocities are given for the inertial frame of reference associated with the cart before colliding with a particle

Maybe I should be more explicit?

 

[haruspex]

Let's look at my first post

Maybe I should be more explicit?

Ok, I missed that.

 

[pbuk]

Let's look at my first post

Maybe I should be more explicit?

I missed that too. You definitely do need to work on your communication skills, but rather than being explicit I think you should focus on expressing things simply and clearly. A little humility wouldn't hurt as well.

Since the problem statement says the cart starts at rest, you are in fact using the ground frame, so why not say so?

Exactly.

 

[MaratZakirov]

Hello everyone, I must admit that I recently failed to find exact solution for equations I wrote above. I can of course make a brute-force or even gradient based solution, but exact analytical solution I can not, I am already spent too much time for it. So if you have some ready to show solution, I'll be very thankful for that.

 

[haruspex]

Hello everyone, I must admit that I recently failed to find exact solution for equations I wrote above. I can of course make a brute-force or even gradient based solution, but exact analytical solution I can not, I am already spent too much time for it. So if you have some ready to show solution, I'll be very thankful for that.

I find it easier to drop the vector form for a problem like this. I set the angle of the sail to rail as theta, incoming particle to sail as phi, departing angle to sail as psi, incoming speed u, departing speed u', cart's acquired speed v.
I got a rather messy quadratic in x where $x=\frac{u'^2}{u^2}$.

Please post your working as far as you get.