My physics textbook asks me to prove that when a billiard ball moving along the x-axis hits another billiard ball of equal mass in an elastic, non-head on, collision, the two balls will move away from each other at a 90 degree angle (see attached picture).
I understand their explanation except for one detail: the analysis - as one would expect - starts with the formulae for momentum for both the $x$- and the $y$-ingredients of the balls' motion. For the one dealing with the y-ingredient, they give: $v_1\sin\Theta_1-v_2\sin\Theta_2=0$.
This is of course based on the fact that we defined the x-axis according to the initial movement of the moving ball, which means that there was no y-component of momentum, so the y-component needs to total 0 after the collision as well.
My problem is the following: as a formula I learnt that $m\mathbf{V}_1 + m\mathbf{V}_2 = m\mathbf{v}_1 + m\mathbf{v}_2$, not $m\mathbf{V}_1 + m\mathbf{V}_2 = m\mathbf{v}_1 - m\mathbf{v}_2$.
I assume that the minus in the answer is there because we know that the velocity of the second ball along the y-axis is in the opposite direction of the other ball.
But that leads to another oddity, namely that if I rearrange the formula, I get: $v_1\sin\Theta_1=v_2\sin\Theta_2$ rather than $v_1\sin\Theta_1=-v_2\sin\Theta_2$.
So the velocities end up being identical, despite the fact that they are going in opposite directions…
Could somebody clear up the confusion?
Thank you!
The answer: