What is the law of conservation of momentum?

The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In a collision between two objects, the total momentum before the collision equals the total momentum after: m1*v1i + m2*v2i = m1*v1f + m2*v2f. Momentum is a vector quantity (it has both magnitude and direction), so signs matter: velocities to the right are positive and velocities to the left are negative by convention.

What is the difference between an elastic and an inelastic collision?

In an elastic collision, both momentum and kinetic energy are conserved. The objects bounce off each other without any energy being converted to heat, sound, or deformation. In a perfectly inelastic collision, momentum is conserved but kinetic energy is not: the objects stick together after impact and move as one combined mass. The maximum kinetic energy is lost in a perfectly inelastic collision. Real-world collisions fall somewhere between these two extremes.

How do you find final velocities in an elastic collision?

For an elastic collision between two objects, you solve the simultaneous equations for conservation of momentum and conservation of kinetic energy. The standard result is: v1f = ((m1 - m2) * v1i + 2 * m2 * v2i) / (m1 + m2) and v2f = ((m2 - m1) * v2i + 2 * m1 * v1i) / (m1 + m2). For a perfectly inelastic collision where the objects stick together, the combined final velocity is: vf = (m1 * v1i + m2 * v2i) / (m1 + m2).

Conservation of Momentum Calculator

This calculator solves one-dimensional collision problems using the law of conservation of momentum, the physics principle stating that the total momentum of two colliding objects stays the same before and after impact, provided no external force interferes. It is built for physics students and anyone checking collision working by hand, whether for homework, a lab report, or general curiosity about how objects behave when they strike each other. You enter the mass and initial velocity of each object, using a positive value for motion to the right and a negative value for motion to the left, then choose whether the collision is elastic (objects bounce apart with kinetic energy conserved) or perfectly inelastic (objects stick together and move as one combined mass). The calculator returns the final velocity of each object, the total momentum before and after the collision, and the kinetic energy before and after, together with how much energy was lost or gained. A breakdown table shows the momentum and kinetic energy contributed by each object individually, and a plain-language summary explains what happened in the collision. Use it to check textbook answers, explore how changing mass or speed affects the outcome, or understand the practical difference between elastic and inelastic collisions. It models an idealised, frictionless, one-dimensional system, so real-world collisions involving rotation, deformation, or motion in more than one direction will differ from these results.

Standard physics formula Law of conservation of momentum; elastic and perfectly inelastic collision equations.

1. Object 1

Mass of object 1 (kg)

Initial velocity of object 1 (m/s, + = right)

m/s

2. Object 2

Mass of object 2 (kg)

Initial velocity of object 2 (m/s, + = right)

m/s

Collision type

Please enter valid numbers for all fields. Masses must be greater than zero.

Collision Results

Final velocity (obj 1)
-0.5 m/s
m/s after collision

Final velocity (obj 2)
5.5 m/s
m/s after collision

Total momentum

14 kg m/s

kg m/s (conserved)

Kinetic energy lost

0 J

joules

Momentum Breakdown

Momentum of object 1 (before)+20 kg m/s

Momentum of object 2 (before)-6 kg m/s

Total momentum (before)+14 kg m/s

Momentum of object 1 (after)-2.5 kg m/s

Momentum of object 2 (after)+16.5 kg m/s

Total momentum (after)+14 kg m/s

Kinetic Energy Breakdown

KE of object 1 (before)40 J

KE of object 2 (before)6 J

Total KE (before)46 J

KE of object 1 (after)0.625 J

KE of object 2 (after)45.375 J

Total KE (after)46 J

Energy change0 J (conserved)

Summary: Enter values above to see results.

How Conservation of Momentum Works

The law of conservation of momentum is a fundamental principle of classical physics. It states that in a closed system (one with no external forces such as friction or gravity acting along the direction of motion), the total momentum before a collision equals the total momentum after the collision.

Momentum (p) is the product of mass and velocity:

p = m × v

where m is mass in kilograms and v is velocity in metres per second. Because velocity has direction, momentum is a vector. Velocities (and momenta) to the right are positive by convention; velocities to the left are negative.

Collision Types Explained

Collision Type	Momentum	Kinetic Energy	What Happens
Elastic	Conserved	Conserved	Objects bounce off; no energy lost to heat or deformation
Perfectly inelastic	Conserved	Not conserved (maximum loss)	Objects stick together and move as one mass
Inelastic (general)	Conserved	Partially lost	Objects bounce off but some energy is dissipated

This calculator handles elastic and perfectly inelastic collisions. Real-world collisions (such as a car crash or a ball bouncing on a floor) are typically partially inelastic, falling between the two extremes.

Formulas Used

Elastic collision (momentum and kinetic energy both conserved). Solving the two simultaneous equations gives the standard result:

v1f = ((m1 - m2) × v1i + 2 × m2 × v2i) / (m1 + m2)
v2f = ((m2 - m1) × v2i + 2 × m1 × v1i) / (m1 + m2)

Perfectly inelastic collision (objects stick together; only momentum conserved). Both objects move together with the same final velocity:

vf = (m1 × v1i + m2 × v2i) / (m1 + m2)

Kinetic energy:

KE = ½ × m × v²

Worked Example

Object 1: mass 5 kg, moving right at 4 m/s. Object 2: mass 3 kg, moving left at 2 m/s (v2i = -2 m/s). Elastic collision.

Total momentum before: (5 × 4) + (3 × -2) = 20 - 6 = 14 kg m/s
v1f = ((5 - 3) × 4 + 2 × 3 × -2) / (5 + 3) = (8 - 12) / 8 = -4/8 = -0.50 m/s (now moving left)
v2f = ((3 - 5) × -2 + 2 × 5 × 4) / (5 + 3) = (4 + 40) / 8 = 44/8 = 5.50 m/s (moving right)
Total momentum after: (5 × -0.5) + (3 × 5.5) = -2.5 + 16.5 = 14 kg m/s (conserved)
KE before: 0.5 × 5 × 4² + 0.5 × 3 × 2² = 40 + 6 = 46 J
KE after: 0.5 × 5 × 0.5² + 0.5 × 3 × 5.5² = 0.625 + 45.375 = 46 J (conserved)

Related Calculators

Sources and method: Newton's laws of motion and the law of conservation of momentum (classical mechanics). Elastic collision equations derived from simultaneous solution of conservation of momentum (m1v1i + m2v2i = m1v1f + m2v2f) and conservation of kinetic energy (0.5m1v1i² + 0.5m2v2i² = 0.5m1v1f² + 0.5m2v2f²). Reference: Halliday, Resnick and Krane, Physics (5th ed.), Chapters 9 and 10.

This calculator uses idealised one-dimensional collision models. Real-world collisions may involve friction, deformation, angular momentum, or forces in multiple dimensions not captured here. For accurate engineering calculations, consult a qualified engineer or physicist.