What are collisions ? Explain the possible types of collisions ? Develop the theory of one dimensional elastic collision ?

Collision : The strong interaction among bodies involving exchange of momenta in small interval of time is called collision.
Eg : collision between 2-billiard balls.
There are two types of collisions. They are elastic and inelastic collisions.
Elastic collisions : The collision in which both momentum and kinetic energy are conserved is known as “Elastic collision”.
Eg : Collisions between nuclei and fundamental particles.
Inelastic collisions : The collision in which only momentum is conserved but not kinetic energy is called as “Inelastic collision”.
Eg : Hitting a ball with a bat.
Collision in one dimension (Elastic collision) :
Consider two bodies ‘A’ and ‘B’ of masses ‘m₁’ and ‘m₂’ respectively and are moving along the same straight line in the same direction.
Let u₁, u₂ be their velocities before collision (u₁ > u₂).
After collision their velocities change to v₁ and v₂ respectively and they move along the same line in the same direction.

Applying law of conservation of linear momentum. Total momentum before collision = total momentum after collision.
$\begin{array}{l}{\mathrm{m}}_1{\mathrm{u}}_1+{\mathrm{m}}_2{\mathrm{u}}_2={\mathrm{m}}_1{\mathrm{v}}_1+{\mathrm{m}}_2{\mathrm{v}}_2\\ {\mathrm{m}}_1\left({\mathrm{u}}_1-{\mathrm{v}}_1\right)={\mathrm{m}}_2\left({\mathrm{v}}_2-{\mathrm{u}}_2\right) ...\left(1\right)\end{array}$
Applying law of conservation of kinetic energy
$\begin{array}{l}\frac{1}{2}{\mathrm{m}}_1{\mathrm{u}}_1^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{u}}_2^2=\frac{1}{2}{\mathrm{m}}_1{\mathrm{v}}_1^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}_2^2\\ \frac{1}{2}{\mathrm{m}}_1\left({\mathrm{u}}_1^2-{\mathrm{v}}_1^2\right)=\frac{1}{2}{\mathrm{m}}_2\left({\mathrm{v}}_2^2-{\mathrm{u}}_2^2\right) ... \left(2\right)\end{array}$
Dividing eqn (2) with eqn (1),
we get
$\begin{array}{l}\frac{{\mathrm{m}}_1\left({\mathrm{u}}_1+{\mathrm{v}}_1\right)\left({\mathrm{u}}_1-{\mathrm{v}}_1\right)}{{\mathrm{m}}_1\left({\mathrm{u}}_1-{\mathrm{v}}_1\right)}=\frac{{\mathrm{m}}_2\left({\mathrm{v}}_2+{\mathrm{u}}_2\right)\left({\mathrm{v}}_2-{\mathrm{u}}_2\right)}{{\mathrm{m}}_2\left({\mathrm{v}}_2-{\mathrm{u}}_2\right)}\\ {\mathrm{u}}_1+{\mathrm{v}}_1={\mathrm{v}}_2+{\mathrm{u}}_2\\ \Rightarrow \left({\mathrm{u}}_1-{\mathrm{u}}_2\right)=\left({\mathrm{v}}_2-{\mathrm{v}}_1\right) ...\left(3\right)\end{array}$
The relative velocity of approaching before collision is equal to relative velocity of seperation after collision.
From Eq (3), ${\mathrm{v}}_2={\mathrm{u}}_1+{\mathrm{v}}_1-{\mathrm{u}}_2\text{ Sub. in eq (1) }$
$\begin{array}{l}{\mathrm{m}}_1\left({\mathrm{u}}_1-{\mathrm{v}}_1\right)={\mathrm{m}}_2\left({\mathrm{u}}_1+{\mathrm{v}}_1-{\mathrm{u}}_2-{\mathrm{u}}_2\right)\\ {\mathrm{m}}_1{\mathrm{u}}_1-{\mathrm{m}}_1{\mathrm{v}}_1={\mathrm{m}}_2{\mathrm{u}}_1+{\mathrm{m}}_2{\mathrm{v}}_1-2{\mathrm{m}}_2{\mathrm{u}}_2\\ {\mathrm{v}}_1=\left(\frac{{\mathrm{m}}_1-{\mathrm{m}}_2}{{\mathrm{m}}_1+{\mathrm{m}}_2}\right){\mathrm{u}}_1+\left(\frac{2{\mathrm{m}}_2}{{\mathrm{m}}_1+{\mathrm{m}}_2}\right){\mathrm{u}}_2\end{array}$
From Eq (3) :
$\begin{array}{l}{\mathrm{v}}_1={\mathrm{v}}_2-{\mathrm{u}}_1+{\mathrm{u}}_2\text{ sub. it in Eq (1) }\\ {\mathrm{m}}_1\left({\mathrm{u}}_1-{\mathrm{v}}_2+{\mathrm{u}}_1-{\mathrm{u}}_2\right)={\mathrm{m}}_2\left({\mathrm{v}}_2-{\mathrm{u}}_2\right)\\ {\mathrm{m}}_1{\mathrm{u}}_1-{\mathrm{m}}_1{\mathrm{v}}_2+{\mathrm{m}}_1{\mathrm{u}}_1-{\mathrm{m}}_1{\mathrm{u}}_2={\mathrm{m}}_2{\mathrm{v}}_2-{\mathrm{m}}_2{\mathrm{u}}_2\\ \Rightarrow {\mathrm{m}}_1{\mathrm{u}}_1-{\mathrm{m}}_2{\mathrm{u}}_1+2{\mathrm{m}}_2{\mathrm{u}}_2=\left({\mathrm{m}}_1+{\mathrm{m}}_2\right){\mathrm{v}}_1\\ 2{\mathrm{m}}_1{\mathrm{u}}_1+{\mathrm{m}}_2{\mathrm{u}}_2-{\mathrm{m}}_1{\mathrm{u}}_2={\mathrm{m}}_1{\mathrm{v}}_2+{\mathrm{m}}_2{\mathrm{v}}_2\\ {\mathrm{v}}_2=\left(\frac{2{\mathrm{m}}_1}{{\mathrm{m}}_1+{\mathrm{m}}_2}\right){\mathrm{u}}_1+\left(\frac{{\mathrm{m}}_2-{\mathrm{m}}_1}{{\mathrm{m}}_1+{\mathrm{m}}_2}\right){\mathrm{u}}_2\end{array}$