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Elastic head-on collision velocities
Intermediate
Final velocities follow from momentum and energy conservation.
v₁ = ((m₁−m₂)u₁+2m₂u₂)/(m₁+m₂)From the two conservation laws: v₁ = ((m₁ − m₂)u₁ + 2m₂u₂)/(m₁ + m₂) and similarly for v₂. These show how the mass ratio determines the outcome.
m₁ = 2, m₂ = 1, u₁ = 3, u₂ = 0: v₁ = 1, v₂ = 4 m/s.
Check by confirming relative speed is reversed.
In more detail
In a one-dimensional elastic collision, the velocities after the collision follow from conservation of momentum and kinetic energy. The formulas give each object's velocity from the masses and initial velocities.
Real-world uses
- Physics labs test elastic collision formulas with gliders on air tracks.
- Nuclear physics uses the same formulas to analyse particle scattering.
- Physics students solve for the final velocities of two colliding masses to check that momentum and energy are both conserved.
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Frequently asked questions
Do I need both conservation laws?
Yes. Momentum gives one equation and kinetic energy gives another, and together they fix both final velocities.
Is there a simpler form?
Yes, using the relative velocity, which reverses in sign in an elastic collision.
How many unknowns are there in an elastic collision?
Two final velocities, which can be found from the two conservation equations for momentum and kinetic energy.
Related topics in Momentum & Collisions
Inelastic collision · Perfectly inelastic collision · Coefficient of restitution · Equal masses collide elastically · Light hits heavy object · Heavy hits light object