Derivation of relativistic momentum

Question

I realise that a dozen questions have been asked about this, and I have scoured through these but I cannot seem to find an answer to my question; we derived the relatvistic momentum in lectures and there is one step in the lecturers reasoning that does not make sense to me, and I cannot seem to find a similar derivation.

It started off with the standard set up on two reference frames, in both of which a ball is thrown directly up/down at a speed $u_0$

NOTE: I tried to upload an image here (actually I tried with three different images from the web) and in each case it was coming up with an error. I'm not sure whether this is a problem on my side or a temporary glitch with stack exchange.

From the addition of speeds, you find that the vertical component of the velocity of the ball thrown in the other frame is $-u_0/\gamma$

Therefore the mometum cannot simply be $p=mv$ if it is to be conserved. This is all fine.

But then my lecturer writes that

Assuming conservation of mometum applies, conserving momentum vertically in frame S gives

$-psin\theta +mu_0=mu_0-psin\theta$

$\implies mu_0=psin\theta$

This is driving me absolutely mad. It seems that my lecturer made no assumption as to what the momentum of the particle that is not simply moving vertically up and down is (and instead just labelled the momentum as $p$ with the vertical component $psin\theta$, and then she randomly assumed that the momentum of the ball which only moves vertically with a speed $u_0$ in this frame remains, as it classically would, as $u_0m$!!

Perhaps I am missing something here, but my lecturer gets the usual expression for relativistic momentum in the end, through the following steps

$p=\frac{mu_0}{sin\theta}$

But $sin\theta=\frac{u_0}{\gamma w}$ (where w is the total speed of the particle in this frame)

$\implies u_0=\gamma w sin\theta$

Thus $p=\gamma m w$

So I am just baffled as to why the momentum of the particle which was thrown vertically up/down in that frame did not change from the classical expression.

I apologise again for the lack of diagram- I know it would make the explanation much clearer, but I tried to upload three different images from the net, and a photo from my phone, but neither of these were uploading.

Answer 1

Conservation of vertical momentum:

When you give a vertically moving object some horizontal momentum, the object only gets horizontal momentum. So its vertical momentum is the same as before you pushed it.

If you change from a frame where a ball is moving only vertically to a frame where the ball is moving also horizontally, the principle of relativity requires that the situation in the new frame looks the same as if the ball was pushed horizontally. So the vertical momentum of the ball must not change in a horizontal frame change either.