# Elastic Collisions in One Dimension

## An elastic collision is a collision between two or more bodies in which kinetic energy is conserved.

#### Key Points

• An elastic collision will not occur if kinetic energy is converted into other forms of energy.

• While molecules do not undergo elastic collisions, atoms often undergo elastic collisions when they collide.

• If two particles are involved in an elastic collision, the velocity of the first particle after collision can be expressed as:<equation contenteditable="false">$v_{1f} =\frac{(m_1-m_2)}{(m_2+m_1)}v_{1i}+\frac{2\cdot m_2}{(m_2+m_1)}v_{2i}$.

• If two particles are involved in an elastic collision, the velocity of the second particle after collision can be expressed as:$v_{2f} =\frac{2\cdot m_1}{(m_2+m_1)}v_{1i} +\frac{(m_2-m_1)}{(m_2+m_1)}v_{2i}$.

#### Terms

• An encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter. Elastic collisions occur only if there is no net conversion of kinetic energy into other forms.

• The energy possessed by an object because of its motion, equal to one half the mass of the body times the square of its velocity.

• (of a body in motion) the product of its mass and velocity.

#### Figures

1. ##### Elastic Collision of Two Unequal Masses

In this animation, two unequal masses collide and recoil.

An elastic collision is a collision between two or more bodies in which the total kinetic energy of the bodies before the collision is equal to the total kinetic energy of the bodies after the collision. An elastic collision will not occur if kinetic energy is converted into other forms of energy. It important to understand how elastic collisions work, because atoms often undergo essentially elastic collisions when they collide. On the other hand, molecules do not undergo elastic collisions when they collide . In this atom we will review case of collision between two bodies.

The mathematics of an elastic collision is best demonstrated through an example. Consider a first particle with mass $m_{1}$ and velocity $v_{1i}$ and a second particle with mass $m_{2}$ and velocity $v_{2i}$. If these two particles collide, there must be conservation of momentum before and after the collision. If we know that this is an elastic collision, there must be conservation of kinetic energy by definition. Therefore, the velocities of particles 1 and 2 after the collision ($v_{1f}$ and $v_{2f}$ respectively) will be related to the initial velocities by:

$\frac{1}{2}m_1\cdot v_{1i}^2+\frac{1}{2}m_2\cdot v_{2i}^2=\frac{1}{2}m_1\cdot v_{1f}^2+\frac{1}{2}m_2\cdot v_{2f}^2$ (due to conservation of kinetic energy)

and

$m_1\cdot v_{1i}+m_2\cdot v_{2i}=m_1\cdot v_{1f}+m_2\cdot v_{2f}$ (due to conservation of momentum).

Since we have two equations, we are able to solve for any two unknown variables. In our case, we will solve for the final velocities of the two particles.

By grouping like terms and canceling out the ½ terms, we can rewrite our conservation of kinetic energy equation as:

$m_1\cdot (v_{1i}^2-v_{1f}^2) = m_2\cdot (v_{2f}^2-v_{2i}^2)$. (Eq.1)

By grouping like terms from our conservation of momentum equation we can find:

$m_1\cdot (v_{1i}-v_{1f}) = m_2\cdot (v_{2f}-v_{2i})$. (Eq. 2)

If we then divide Eq. 1 by Eq. 2 and perform some cancelations we will find:

$v_{1i} + v_{1f} = v_{2f} + v_{2i}$. (Eq. 3)

We can solve for $v_{1f}$ as:

$v_{1f} = v_{2f} + v_{2i}-v_{1i}$.  (Eq. 4)

At this point we see that $v_{2f}$ is still an unknown variable. So we can fix this by plugging Eq. 4 into our initial conservation of momentum equation. Our conservation of momentum equation with Eq. 4 substituted in looks like:

$m_1\cdot v_{1i}+m_2\cdot v_{2i}=m_1\cdot(v_{2f} + v_{2i}-v_{1i})+m_2\cdot v_{2f}$.  (Eq.5)

After doing a little bit of algebra on Eq. 5 we find:

$v_{2f} =\frac{2\cdot m_1}{(m_2+m_1)}v_{1i} +\frac{(m_2-m_1)}{(m_2+m_1)}v_{2i}$    (Eq.6)

At this point we have successfully solved for the final velocity of the second particle. We still need to solve for the velocity of the first particle, so let us do that by plugging Eq. 6 into Eq. 4.

$v_{1f} = [\frac{2\cdot m_1}{(m_2+m_1)}v_{1i} +\frac{(m_2-m_1)}{(m_2+m_1)}v_{2i}] + v_{2i}-v_{1i}$.  (Eq. 7)

After performing some algebraic manipulation of Eq. 7, we finally find:

$v_{1f} =\frac{(m_1-m_2)}{(m_2+m_1)}v_{1i}+\frac{2\cdot m_2}{(m_2+m_1)}v_{2i}$(Eq. 8)

#### Key Term Glossary

atom
The smallest possible amount of matter which still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.
##### Appears in these related concepts:
conservation
A particular measurable property of an isolated physical system does not change as the system evolves.
##### Appears in these related concepts:
conservation of momentum
In a closed system the total momentum is constant. $\sum_i \vec{p}_i=\sum_f \vec{p}$
##### Appears in these related concepts:
elastic
Capable of stretching; particularly, capable of stretching so as to return to an original shape or size when force is released.
##### Appears in these related concepts:
elastic collision
An encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter. Elastic collisions occur only if there is no net conversion of kinetic energy into other forms.
##### Appears in these related concepts:
energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
kinetic
Of or relating to motion
##### Appears in these related concepts:
kinetic energy
The energy possessed by an object because of its motion, equal to one half the mass of the body times the square of its velocity.
##### Appears in these related concepts:
Kinetic Energy
The energy associated with a moving particle or object having a certain mass.
##### Appears in these related concepts:
mass
The quantity of matter which a body contains, irrespective of its bulk or volume. It is one of four fundamental properties of matter. It is measured in kilograms in the SI system of measurement.
##### Appears in these related concepts:
momentum
(of a body in motion) the product of its mass and velocity.
##### Appears in these related concepts:
particle
A very small piece of matter, a fragment; especially, the smallest possible part of something.
##### Appears in these related concepts:
velocity
A vector quantity that denotes the rate of change of position with respect to time, or a speed with a directional component.
##### Appears in these related concepts:
Velocity
The rate of change of displacement with respect to change in time.
##### Appears in these related concepts:
work
A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.