Elastic Collisions in One Dimension
An elastic collision is a collision between two or more bodies in which kinetic energy is conserved.
Learning Objective

Assess the relationship among the collision equations to derive elasticity
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:
$v_{1f} =\frac{(m_1m_2)}{(m_2+m_1)}v_{1i}+\frac{2\cdot m_2}{(m_2+m_1)}v_{2i}$ .  If two particles are involved in an elastic collision, the velocity of the second particle after collision can be expressed as:
$v_{2f} =\frac{2\cdot m_1}{(m_2+m_1)}v_{1i} +\frac{(m_2m_1)}{(m_2+m_1)}v_{2i}$ .
Terms

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.

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.

momentum
(of a body in motion) the product of its mass and velocity.
Full Text
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
and
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:
By grouping like terms from our conservation of momentum equation we can find:
If we then divide Eq. 1 by Eq. 2 and perform some cancelations we will find:
We can solve for
At this point we see that
After doing a little bit of algebra on Eq. 5 we find:
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.
After performing some algebraic manipulation of Eq. 7, we finally find:
Elastic Collision of Two Unequal Masses
In this animation, two unequal masses collide and recoil.
Key Term Reference
 atom
 Appears in these related concepts: Overview of Atomic Structure, Description of the Hydrogen Atom, and Stable Isotopes
 conservation
 Appears in these related concepts: Conservation of Mechanical Energy, Museums and Private Collections, and Linear Momentum
 dimension
 Appears in these related concepts: TwoDimensional Space, Length, and Dimensional Analysis
 elastic
 Appears in these related concepts: Defining Price Elasticity of Demand, Applications of Elasticities, and Tax Incidence, Efficiency, and Fairness
 energy
 Appears in these related concepts: Surface Tension, Energy Transportation, and Introduction to Work and Energy
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 kinetic
 Appears in these related concepts: Friction: Static, The Kinetic Molecular Theory of Matter, and Postmodernist Sculpture
 mass
 Appears in these related concepts: Mass Spectrometer, Pop Art, and Mass
 velocity
 Appears in these related concepts: Velocity of Blood Flow, RootMeanSquare Speed, and Rolling Without Slipping
 work
 Appears in these related concepts: Heat and Work, Free Energy and Work, and The First Law of Thermodynamics
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