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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

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.

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.
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:
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Key Term Reference
 atom
 Appears in this related concepts: Description of the Hydrogen Atom, Early Ideas about Atoms, and Periods 1 through 3
 conservation
 Appears in this related concepts: Conservation of Mechanical Energy, Museums and Private Collections, and Linear Momentum
 dimension
 Appears in this related concepts: Length, Elastic Collisions in Multiple Dimensions, and Volume
 elastic
 Appears in this related concepts: Internal vs. External Forces, Applications of Elasticities, and Demand Curve
 energy
 Appears in this related concepts: Surface Tension, Introduction to Work and Energy, and The Role of Energy and Metabolism
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 kinetic
 Appears in this related concepts: Friction: Static, The Kinetic Molecular Theory of Matter, and Sculpture
 mass
 Appears in this related concepts: Physical and Chemical Properties of Matter, Mass Spectrometer, and Mass
 velocity
 Appears in this related concepts: Rolling Without Slipping, Tangent and Velocity Problems, and Applications and ProblemSolving
 work
 Appears in this related concepts: Force at an Angle to Displacement, Conservation of Energy in Rotational Motion, and Heat and Work
Sources
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Cite This Source
Source: Boundless. “Elastic Collisions in One Dimension.” Boundless Physics. Boundless, 06 Jan. 2015. Retrieved 24 Mar. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/linearmomentumandcollisions7/collisions70/elasticcollisionsinonedimension2986220/