Collision Momentum Calculator

Calculate velocities after elastic or inelastic collisions using conservation of momentum.

Negative = opposite direction
1=elastic, 0=perfectly inelastic
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Momentum Conservation Governs Isolated Collisions

During a short collision, total momentum of an isolated system is conserved even though kinetic energy may or may not be conserved. In one dimension, m1u1+m2u2=m1v1+m2v2. Elastic collisions also conserve total kinetic energy, while perfectly inelastic collisions leave the objects moving together and lose the maximum kinetic energy consistent with momentum conservation.

Velocity signs are essential because momentum is a vector. In two or three dimensions, momentum conservation must be applied separately to each component. Kinetic energy that is not conserved in an inelastic collision becomes deformation, heat, sound, and internal energy.

Σpbefore=Σpafter,   perfectly inelastic: v=(m1u1+m2u2)/(m1+m2)
SymbolMeaningWhy it appears / units
mMasskg.
uInitial velocitym/s with sign for direction.
vFinal velocitym/s with sign for direction.
pMomentumkg·m/s; vector quantity.

Momentum conservation comes from negligible external impulse during the collision interval. It does not imply each object keeps its own momentum; objects exchange momentum through large internal forces.

Momentum conservation is a signed check. Compute m1v1+m2v2 before and after using the same positive direction. For a perfectly inelastic collision both objects share one final velocity, which must lie between the initial velocities when the masses are positive.

Worked Examples

Example 1: Elastic: m₁=2kg, u₁=5m/s hits m₂=2kg at rest
Equal masses elastic
Result: v₁=0, v₂=5m/s — velocity transfer
Classic pool ball collision
Example 2: Inelastic: m₁=3kg, u₁=4m/s hits m₂=2kg at rest
p=12, vf=12/5
Result: v=2.4 m/s — KE lost=8.4 J
Real-world crash: energy to deformation
Example 3: Perfectly inelastic collision
m1=2kg at 4m/s, m2=2kg at rest
Result: v=2m/s together
Momentum is conserved while kinetic energy decreases.
Example 4: Equal-mass elastic collision
one mass moving, equal target mass at rest
Result: ideal head-on speeds exchange
The incoming object can stop while the target leaves with its speed.

Common Mistakes

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Conserving kinetic energy in every collision

Only elastic collisions conserve total kinetic energy. Momentum conservation is more general for isolated collision systems.

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Dropping velocity signs

Opposite directions must carry opposite signs in one-dimensional momentum equations.

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Applying momentum conservation when external impulse is important

If a large external force acts over the collision interval, system momentum changes by that external impulse.

Frequently Asked Questions

What is coefficient of restitution?
e = (relative speed after)/(relative speed before) = (v₂-v₁)/(u₁-u₂). e=1: perfectly elastic. e=0: perfectly inelastic. Tennis ball on concrete: e≈0.75. Steel ball on steel: e≈0.95.
When is momentum NOT conserved?
Momentum is conserved only in isolated systems (no external forces). With friction, gravity component, or other external impulses, total momentum changes. In collisions where impact time is short, external impulses are negligible.
Why is momentum conserved when kinetic energy is not?
Internal collision forces occur in equal and opposite pairs, so they cancel in total momentum. Kinetic energy can transform into internal energy, deformation, heat, or sound.
What is coefficient of restitution?
It is the ratio of relative separation speed to relative approach speed along the impact line. A value of 1 is perfectly elastic and 0 is perfectly inelastic along that line.
Can one object gain kinetic energy in a collision?
Yes. Individual objects can gain or lose kinetic energy. The classification depends on the total kinetic energy of the complete system.
How are two-dimensional collisions solved?
Apply momentum conservation independently to x and y components, then use additional information such as elasticity, angles, or restitution to obtain enough equations for the unknowns.
How should velocity signs be handled in a one-dimensional collision?
Choose one direction as positive before substituting numbers. A cart moving the opposite way must have a negative velocity; using all positive speeds destroys the vector information in momentum. After solving, compare total momentum before and after. Kinetic energy is conserved only for an elastic collision, not for every momentum-conserving collision.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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