You could have seen the rubber ball bouncing back, a spring bouncing when given a push, a child bouncing on jumping mat etc. In these bouncing bodies there is no loss in energy nor momentum and we call these as elastic bodies as the collisions they undergo are elastic. An elastic collision is that where both momentum and kinetic energy are
conserved. It implies that no dissipative force acts during the
collision and that the kinetic energy of the objects before collision is
still in the form of kinetic energy after collision. Here there is no change in internal energy of the interacting objects |

The collision where there the kinetic energy is same before and after the collision is what we call perfectly elastic. Here no energy is dissipated as heat. Let illustrate it:

Let v

The velocity after collision vLet v

_{1}be the mass of first body m_{1}and v_{2}be the velocity of mass of second body m_{2}._{1}' and v

_{2}' for perfectly elastic collision is given by

v

_{1}' = $\frac{(m_1 - m_2)\epsilon}{m_1 + m_2}$ v

_{1}

= $\frac{m_1 - m_2}{m_1 + m_2}$ v

_{1}

v

_{2}' = $\frac{m_1 (1 + \epsilon)}{m_1 + m_2}$ v

_{1}

= $\frac{2 m_1}{m_1 + m_2}$ v

_{1}

A elastic collision is one where there is no loss of kinetic energy in the collision Here linear momentum, kinetic energy and total energy is conserved
whereas the mechanical energy and interaction force not conserved. An inelastic collision is that where the kinetic energy is changed into some other form of energy in the collision.

To justify whether the body is elastic or not we use Coefficient of restitution. It measures the degree of elasticity. It is the ratio of relative velocity after collision to its relative velocity before collision. It is given by

To justify whether the body is elastic or not we use Coefficient of restitution. It measures the degree of elasticity. It is the ratio of relative velocity after collision to its relative velocity before collision. It is given by

e = $\frac{relative\vel\after\collision}{relative\velocity\before\collision}$If u

_{1 }and u_{2}are initial velocity and v_{1}and v_{2}are final velocities of the body undergoing collision thene = $\frac{v_2 - v_1}{u_1 - u_2}$

For elastic bodies the coefficient of restitution e = 1 and for inelastic bodies it is less than 1.Take two objects with mass m

Also the principle of kinetic energy is said to conserved

$\frac{1}{2}$ m

Solving the above equations to get v

v

v

_{1}and m_{2}. Let the initial velocity of object 1 be u_{1 }and initial velocity of object 2 be u_{2}and v_{1}and v_{2}be the final velocities respectively. The law of conservation of momentum tells that**m**

_{1}u_{1}+ m_{2}u_{2}= m_{1}v_{1}+ m_{2}v_{2}........(1)$\frac{1}{2}$ m

_{1}u_{1}^{2}+ $\frac{1}{2}$ m_{2}u_{2}^{2}= $\frac{1}{2}$ m_{1}v_{1}^{2}+ $\frac{1}{2}$ m_{2}v_{2}^{2}...........(2)Solving the above equations to get v

_{1}and v_{2}v

_{1}= $\frac{u_{1} (m_{1} - m_{2})+ 2m_{2} u_{2}}{m_{1} + m_{2}}$.....................(3)v

_{2}= $\frac{u_{2} (m_{2} - m_{1})+ 2m_{1} u_{1}}{m_{1} + m_{2}}$.....................(4)If there is no collision then v_{1}= u_{1}and v_{2}= u_{2}then the equation will be v_{1}– v_{2}= u_{2}– u_{1}.The law of conservation of momentum tells that momentum before collision is equal to momentum after collision.

**m**Initial velocity of first body = u

_{1}u_{1}+ m_{2}u_{2}= m_{1}v_{1}+ m_{2}v_{2}........(1)_{1},

Initial velocity of second body = u

Final Velocity of first body = v

Final Velocity of second body = v

If we assume that m_{2},Final Velocity of first body = v

_{1},Final Velocity of second body = v

_{2}._{1}>> m

_{2}then using above relation v

_{1}= u

_{1}and v

_{2}= u

_{2}

One dimensional elastic collision is that where the colliding bodies move along the same straight line path before and after the collision. Consider two bodies A and B of masses m

_{1}and m_{2}moving along the same straight line in the same direction. Let**v**_{1i}and**v**_{2i}be the velocities such that**|$\arrow{v_{1i}}$| > |$\arrow{v_2i}$|**

**$\arrow{v_{1f}}$**and

**$\arrow{v_2f}$**be the velocities of

**A**and

**B**respectively after the collision

Total momentum before collision = Total momentum after collision

**m**

m

_{1}v_{1i}+ m_{2}v_{2i}= m_{1}v_{1f}+ m_{2}v_{2f}m

_{1}(v_{1i}- v_{1f}) = m_{2}(v_{2f}- v_{2i})Kinetic energy before Collision = Kinetic energy after Collision

$\frac{1}{2}$ m

_{1}v

_{1i}

^{2}+ $\frac{1}{2}$ m

_{2}v

_{2i}

^{2}= $\frac{1}{2}$ m

_{1}v

_{1f}

^{2 }+ $\frac{1}{2}$ m

_{2}v

_{2f}

^{2}

Substituting and evaluating the above equations

The one dimensional elastic collisions is given as

v

_{1}= $\frac{m_1 - m_2}{m_1 +m_2}$ u

_{1}+ $\frac{2m_2}{m_1 + m_2}$ u

_{2}

v

_{2}= $\frac{2m_1}{m_1 + m_2}$ u

_{1}+ $\frac{m_2 - m_1}{m_1 + m_2}$ u

_{2}

Here v

_{1}and v

_{2}are the velocities of the bodies after collision.

Momentum before Collision ($\sum$ mv

_{x}) = Momentum after Collision ($\sum$ mv

_{x})

Momentum before Collision ($\sum$ mv

_{y}) = Momentum after Collision ($\sum$ mv

_{y})

Kinetic energy is conserved in elastic collisions

$\sum$ $\frac{1}{2}$ mv

^{2}= $\sum$ $\frac{1}{2}$ mv^{2}The two dimensional elastic collisions is given as

v

_{1x}= $\frac{m_1 - m_2}{m_1 +m_2}$ u_{1x}+ $\frac{2m_2}{m_1 + m_2}$ u_{2x}v

v

_{2x}= $\frac{2m_1}{m_1 + m_2}$ u_{1x}+ $\frac{m_2 - m_1}{m_1 + m_2}$ u_{2x}v

_{1y}= $\frac{m_1 - m_2}{m_1 +m_2}$ u_{1y}+ $\frac{2m_2}{m_1 + m_2}$ u_{2y}v

_{2y}= $\frac{2m_1}{m_1 + m_2}$ u_{1y}+ $\frac{m_2 - m_1}{m_1 + m_2}$ u_{2y}Here v

Here are some common illustrations where you could see elastic collisions_{1}and v_{2}are the velocities of the bodies after collision in x and y coordinates.- You throw a rubber ball on the floor, it bounces back. Here both momentum and kinetic energy are conserved.
- The collision between the atoms is theoretical example of elastic collision. As we don't actually see it
- The collision between two billiard balls is a practical example stating about the elastic collision.