There are two schemes for collsion, which are hard-sphere and soft-sphere.

hard-sphere scheme

In collision with coefficient of restitution $e$

$$ \tag{1} v_{1f} = \frac{ m_1 v_{1i} + m_2 v_{2i} + m_2 \ e (v_{2i} - v_{1i}) }{ m_1 + m_2 } $$

gives final velocity of mass $m_1$ and

$$ \tag{2} v_{2f} = \frac{ m_1 v_{1i} + m_2 v_{2i} + m_1 \ e (v_{1i} - v_{2i}) }{ m_1 + m_2 } $$

gives final velocity of mass $m_2$.

Colission happens when condition

$$ \tag{3} | x_1 - x_2 | < R_1 + R_2 $$

meets.

Without collision

$$ \tag{4} x_1(t + \Delta t) = x_1(t) + v_1(t) \Delta t $$

gives position of mass $m_1$ and

$$ \tag{5} x_2(t + \Delta t) = x_2(t) + v_2(t) \Delta t $$

Before collsion happens

$$ \tag{6} v_{1i} = v_1(t) $$

and

$$ \tag{7} v_{2i} = v_2(t), $$

then after collsion with time interval $\Delta t$

$$ \tag{8} v_1(t + \Delta t) = v_{1f} $$

and

$$ \tag{9} v_2(t + \Delta t) = v_{2f}, $$

where final velocities are obtained using Eqns (1) and (2).

soft-sphere scheme

Overlap between two sphereical mass $m_1$ and mass $m_2$ is given by

$$ \tag{10} \xi_{12} = \max(0, R_1 + R_2 - |x_1 - x_2|) $$

where each has raidius of $R_1$ and $R_2$ and

$$ \tag{11} \dot{\xi}_{12} = -|v_1 - v_2| \ {\rm sign} (\xi _{12}) $$

is its derivative.

Reduce mass of the system is

$$ \tag{12} \mu = \frac{m_1 m_2}{m_1 + m_2}. $$

Linear spring-dashpot mode gives

$$\tag{13} F_{12} = \left( k \xi_ {12} - \gamma \dot\xi_{12} \right) \frac{(x_1 - x_2)}{|x_1 - x_2|}. $$

The 3rd Newton’s law gives

$$\tag{14} F_{21} = - F_{12}. $$

The 2nd Newton’s law allows

$$ \tag{15} a_1 = \frac{1}{\mu} F_{12} $$

and

$$ \tag{16} a_2 = \frac{1}{\mu} F_{21}. $$

Using Euler algorithm

$$ \tag{17} v_1(t + \Delta t) = v_1(t) + a_1 \Delta t $$

and

$$ \tag{18} v_2(t + \Delta t) = v_2(t) + a_2 \Delta t. $$

Then also

$$ \tag{19} x_1(t + \Delta t) = x_1(t) + v_1(T) \Delta t $$

and

$$ \tag{20} x_2(t + \Delta t) = x_2(t) + v_2(t) \Delta t. $$