I'm confused over the Wiki equation for an elastic collision. Does anyone know how equations (1) and (2) are formed to result in (3)? I think I'm overlooking some simple algebra.
Consider particles 1 and 2 with masses $m_1$, $m_2$, and velocities $u_1$, $u_2$ before collision, $v_1$, $v_2$ after collision. The conservation of the total momentum before and after the collision is expressed by:
$$m_{1}u_{1}+m_{2}u_{2} \ =\ m_{1}v_{1} + m_{2}v_{2} \tag 1$$
Likewise, the conservation of the total kinetic energy is expressed by:
$$\tfrac12 m_1u_1^2+\tfrac12 m_2u_2^2 \ =\ \tfrac12 m_1v_1^2 +\tfrac12 m_2v_2^2 \tag 2$$
These equations may be solved directly to find $v_1,v_2$ when $u_1,u_2$ are known.
$$
\begin{array}{ccc}
v_1 &=& \dfrac{m_1-m_2}{m_1+m_2} u_1 + \dfrac{2m_2}{m_1+m_2} u_2 \\[.5em]
v_2 &=& \dfrac{2m_1}{m_1+m_2} u_1 + \dfrac{m_2-m_1}{m_1+m_2} u_2
\end{array} \tag 3
$$
Best Answer
Note that the following procedure is valid only for the one-dimensional elastic collision. If elastic collision happens in two dimensions, then conservation of momentum (your Eq. (1)) gives 2 equations while conservation of energy (your Eq. (2)) gives only 1 equation, in which case you have 3 equations and 4 unknowns (x and y components for two velocities). This means that in the two-dimensional elastic collision you need to know at least one magnitude or one angle for one of the two velocities after the collision in order to be able to solve for other unknowns.
One-dimensional elastic collision
From your Eq. (1) it follows
$$m_1 (u_1 - v_1) = -m_2 (u_2 - v_2) \tag {1a}$$
From your Eq. (2) it follows
$$\frac{1}{2} m_1 (u_1^2 - v_1^2) = -\frac{1}{2} m_2 (u_2^2 - v_2^2)$$
$$m_1 (u_1 - v_1) (u_1 + v_1) = -m_2 (u_2 - v_2) (u_2 + v_2) \tag{2a}$$
From my Eqs. (1a) and (2a) it follows
$$u_1 + v_1 = u_2 + v_2$$
$$\boxed{u_1 - u_2 = -(v_1 - v_2)} \tag {3a}$$
This equation tells you that relative velocity before and after elastic collision has the same magnitude and opposite direction, which is important property of elastic collisions.
You can now combine my Eqs. (1a) and (3a) to reach your Eq. (3)
$$v_2 = u_1 - u_2 + v_1 \qquad \text{and} \qquad v_2 = u_2 + \frac{m_1}{m_2} (u_1 - v_1) \tag {4a}$$
$$\text{give} \qquad v_1 = \frac{m_1 - m_2}{m_1 + m_2} u_1 + \frac{2 m_2}{m_1 + m_2} u_2$$
Procedure is similar to get the expression for velocity $v_2$.