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Elastic Collision


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

Perfectly Elastic Collision

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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 v1 be the mass of first body m1 and v2 be the velocity of mass of second body m2.
The velocity after collision v1' and v2' for perfectly elastic collision is given by

v1' = $\frac{(m_1 - m_2)\epsilon}{m_1 + m_2}$ v1
= $\frac{m_1 - m_2}{m_1 + m_2}$ v1

v2' = $\frac{m_1 (1 + \epsilon)}{m_1 + m_2}$ v1
= $\frac{2 m_1}{m_1 + m_2}$ v1

Elastic and Inelastic Collision

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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
e = $\frac{relative\vel\after\collision}{relative\velocity\before\collision}$If u1 and u2 are initial velocity and v1 and v2 are final velocities of the body undergoing collision then
e = $\frac{v_2 - v_1}{u_1 - u_2}$
For elastic bodies the coefficient of restitution e = 1 and for inelastic bodies it is less than 1.

Elastic Collision Equation

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Take two objects with mass m1 and m2. Let the initial velocity of object 1 be u1 and initial velocity of object 2 be u2 and v1 and v2 be the final velocities respectively. The law of conservation of momentum tells that
m1 u1 + m2 u2 = m1 v1 + m2 v2........(1)
Also the principle of kinetic energy is said to conserved
$\frac{1}{2}$ m1 u12 + $\frac{1}{2}$ m2 u22 = $\frac{1}{2}$ m1 v12+ $\frac{1}{2}$ m2 v22 ...........(2)

Solving the above equations to get v1 and v2
v1 = $\frac{u_{1} (m_{1} - m_{2})+ 2m_{2} u_{2}}{m_{1} + m_{2}}$.....................(3)

v2 = $\frac{u_{2} (m_{2} - m_{1})+ 2m_{1} u_{1}}{m_{1} + m_{2}}$.....................(4)
If there is no collision then v1 = u1 and v2 = u2 then the equation will be v1 – v2 = u2 – u1.

Elastic Collision Momentum

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The law of conservation of momentum tells that momentum before collision is equal to momentum after collision.
m1 u1 + m2 u2 = m1 v1 + m2 v2........(1)Initial velocity of first body = u1,
Initial velocity of second body = u2,
Final Velocity of first body = v1,
Final Velocity of second body = v2.
If we assume that m1 >> m2 then using above relation v1 = u1 and v2 = u2

Elastic Collision in One dimension

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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 m1 and m2 moving along the same straight line in the same direction. Let v1i and v2i be the velocities such that
|$\arrow{v_{1i}}$| > |$\arrow{v_2i}$|
Let $\arrow{v_{1f}}$ and $\arrow{v_2f}$ be the velocities of A and B respectively after the collision

Applying the law of conservation of momentum
Total momentum before collision = Total momentum after collision
m1v1i + m2v2i = m1v1f + m2v2f
m1(v1i - v1f) = m2 (v2f - v2i)

Kinetic energy before Collision = Kinetic energy after Collision
$\frac{1}{2}$ m1v1i2 + $\frac{1}{2}$ m2v2i2 = $\frac{1}{2}$ m1 v1f2 + $\frac{1}{2}$ m2 v2f2
Substituting and evaluating the above equations
The one dimensional elastic collisions is given as

v1 = $\frac{m_1 - m_2}{m_1 +m_2}$ u1 + $\frac{2m_2}{m_1 + m_2}$ u2

v2 = $\frac{2m_1}{m_1 + m_2}$ u1 + $\frac{m_2 - m_1}{m_1 + m_2}$ u2
Here v1 and v2 are the velocities of the bodies after collision.

Two Dimensional Elastic Collision

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In elastic collision two things to be noted i.e., Momentum and kinetic energy. Momentum is conserved in each dimension
Momentum before Collision ($\sum$ mvx) = Momentum after Collision ($\sum$ mvx)
Momentum before Collision ($\sum$ mvy) = Momentum after Collision ($\sum$ mvy)

Kinetic energy is conserved in elastic collisions
$\sum$ $\frac{1}{2}$ mv2 = $\sum$ $\frac{1}{2}$ mv2

The two dimensional elastic collisions is given as
v1x = $\frac{m_1 - m_2}{m_1 +m_2}$ u1x + $\frac{2m_2}{m_1 + m_2}$ u2x

v2x = $\frac{2m_1}{m_1 + m_2}$ u1x + $\frac{m_2 - m_1}{m_1 + m_2}$ u2x

v1y = $\frac{m_1 - m_2}{m_1 +m_2}$ u1y + $\frac{2m_2}{m_1 + m_2}$ u2y

v2y = $\frac{2m_1}{m_1 + m_2}$ u1y + $\frac{m_2 - m_1}{m_1 + m_2}$ u2y
Here v1 and v2 are the velocities of the bodies after collision in x and y coordinates.

Elastic Collision Examples

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Here are some common illustrations where you could see elastic collisions
  • 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.