A basic formula of 1D elastic collision derivation I found the folowing formula for 1d elastic colition:
$$v_1 +u_1 =v_2+u_2 $$
(v,u means the velocity before and after the collision respectively)
I tried to derivate it from momentum and energy conservations but didn't see how it works. Moreover that, when I tried to see if it works it seems to work for mass ratio of 1,2 and infinity (wall colision).
A problem I saw was that obviously when there is no collision the law dous not apply. Can anyone help me pinpoint what I miss? 
 A: Conservation of momentum and kinetic energy:
$$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$$
$$\frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2$$
can be rewritten to:
$$m_1\left(v_1 - u_1\right) = -m_2\left(v_2 - u_2\right)$$
$$m_1\left(v_1^2 - u_1^2\right) = -m_2\left(v_2^2 - u_2^2\right)$$
Substituting: $v_i^2 - u_j^2 = \left(v_i+u_j\right)\left(v_i-u_j\right)$
yields: 
$$ m_1\left(v_1 - u_1\right) = - m_2\left(v_2 - u_2\right)$$
$$m_1\left(v_1+u_1\right)\left(v_1-u_1\right) = -m_2\left(v_2+u_2\right)\left(v_2-u_2\right)$$
Dividing the latter by the former equation yields:
$$ v_1 + u_1 = v_2 + u_2$$
which is the equation you are looking for
A: here's the proof.
Let $$KE_1 = \frac{1}{2}m_1(v_1^2 - u_1^2)$$
Let $$KE_2 = \frac{1}{2} m_2(v_2^2 - u_2^2)$$
Since it is an elastic collision, KE must be conserved.
$$KE_1 - KE_2 = 0$$ 
$$KE_1 = KE_2$$
$$\frac{1}{2}m_1(v_1^2 - u_1^2) = \frac{1}{2}m_2(v_2^2 - u_2^2)$$
$$m_1 = m_2.\frac{v_2^2 - u_2^2}{v_1^2 - u_1^2}$$
By conservation of momentum,
$$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$$
Substituting for $$m_1$$,
$$m_2.\frac{v_2^2 - u_2^2}{v_1^2 - u_1^2}u_1 + m_2u_2 = m_2.\frac{v_2^2 - u_2^2}{v_1^2 - u_1^2}v_1 + m_2v_2$$
Cancel $$m_2$$,
$$\frac{v_2^2 - u_2^2}{v_1^2 - u_1^2}u_1 + u_2 = \frac{v_2^2 - u_2^2}{v_1^2 - u_1^2}v_1 + v_2$$
$$\frac{v_1-u_1}{u_2-v_2}.\frac{(v_2 + u_2)(v_2 - u_2)}{(v_1 + u_1)(v_1 - u_1)} = 1$$
$$\frac{(-1)(u_2 + v_2)}{v_1 + u_1} = 1$$
$$(-1)(u_2 + v_2) = v_1 + u_1$$
$$|u_2 + v_2| = |v_1 + u_1|$$
Let me know if you need to clarify any part of the proof.
