About

Momentum conservation governs every collision in classical mechanics. A miscalculated final velocity in vehicle crash reconstruction can invalidate an entire forensic report. In engineering, incorrect impulse estimates lead to under-designed bumpers or over-specified dampeners. This calculator solves both elastic and perfectly inelastic one-dimensional collisions using the closed-form equations derived from m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′. It reports final velocities, kinetic energy before and after impact, energy dissipated as heat or deformation, and the coefficient of restitution e. The model assumes point masses in one dimension with no external forces during the collision interval. Real collisions involve friction, rotational inertia, and material deformation not captured here.

Formulas

The law of conservation of linear momentum states that the total momentum of an isolated system remains constant before and after collision.

m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′

For a perfectly elastic collision, kinetic energy is also conserved. The closed-form final velocities are:

v₁′ = (m₁ − m₂)v₁ + 2m₂v₂m₁ + m₂

v₂′ = (m₂ − m₁)v₂ + 2m₁v₁m₁ + m₂

For a perfectly inelastic collision, the objects stick together. The combined final velocity is:

v_f = m₁v₁ + m₂v₂m₁ + m₂

Kinetic energy before and after is computed as:

KE = 12mv²

The coefficient of restitution quantifies the elasticity of the collision:

e = −v₁′ − v₂′v₁ − v₂

Where m₁, m₂ are the masses of objects 1 and 2 in kg. v₁, v₂ are initial velocities in m/s. v₁′, v₂′ are final velocities. e = 1 for perfectly elastic, e = 0 for perfectly inelastic.

Reference Data

Scenario	Object 1	Mass m₁	v₁	Object 2	Mass m₂	v₂	Type	Restitution e
Billiard balls	Cue ball	0.17 kg	2.0 m/s	Object ball	0.17 kg	0 m/s	Near-elastic	0.95
Newton's cradle	Steel sphere	0.10 kg	1.5 m/s	Steel sphere	0.10 kg	0 m/s	Elastic	1.00
Car rear-end crash	Sedan	1500 kg	15 m/s	SUV (stopped)	2200 kg	0 m/s	Inelastic	0.15
Football tackle	Linebacker	110 kg	7 m/s	Receiver	85 kg	−3 m/s	Perfectly inelastic	0.00
Freight train coupling	Locomotive	90000 kg	5 m/s	Freight car	30000 kg	0 m/s	Perfectly inelastic	0.00
Proton-proton (LHC)	Proton	1.67×10⁻²⁷ kg	3×10⁷ m/s	Proton (rest)	1.67×10⁻²⁷ kg	0 m/s	Elastic	1.00
Tennis serve & return	Racket	0.33 kg	40 m/s	Tennis ball	0.058 kg	−20 m/s	Near-elastic	0.75
Ice hockey puck pass	Stick+puck	0.90 kg	30 m/s	Goalie pad	5.0 kg	0 m/s	Inelastic	0.30
Ballistic pendulum	Bullet	0.01 kg	400 m/s	Wood block	2.5 kg	0 m/s	Perfectly inelastic	0.00
Spacecraft docking	Capsule	8000 kg	0.1 m/s	Station module	20000 kg	0 m/s	Perfectly inelastic	0.00
Bumper cars	Car A	300 kg	3 m/s	Car B	280 kg	−2 m/s	Near-elastic	0.80
Neutron moderation	Neutron	1.675×10⁻²⁷ kg	2×10⁶ m/s	Hydrogen nucleus	1.673×10⁻²⁷ kg	0 m/s	Elastic	1.00

Frequently Asked Questions

When m₁ = m₂, the elastic collision formulas simplify to a complete velocity exchange: v₁′ = v₂ and v₂′ = v₁. This is the principle behind Newton's cradle. If one object is at rest, the moving object stops completely and transfers all its momentum.

The fraction of kinetic energy lost equals m₁m₂(v₁ − v₂)²2(m₁ + m₂). This energy is converted to heat, sound, and permanent deformation. For two equal masses colliding head-on at the same speed, exactly 100% of kinetic energy is lost.

Yes. This calculator uses a 1D axis where positive velocity means motion to the right and negative means motion to the left. A head-on collision between two objects requires opposite signs. If both velocities are positive, object 2 is being overtaken from behind. The sign of the result tells you the direction of travel after impact.

Strictly, no. The law holds only for isolated systems with zero net external force during the collision interval. In practice, collisions happen over such short time intervals (milliseconds) that external forces like gravity and friction produce negligible impulse compared to the collision forces. This is the impulse approximation used in forensic accident reconstruction.

When v₁ = v₂, the denominator (v₁ − v₂) is zero. Physically, no collision occurs because the objects have zero relative approach velocity. The calculator flags this as an invalid scenario.

Energy loss fraction approaches 2m_smallm_large as the mass ratio grows. A bullet embedding in a heavy block (ballistic pendulum) loses over 99% of kinetic energy while still conserving momentum perfectly. This is why ballistic pendulums measure momentum, not energy.