Conservation of Momentum: 3 Conditions, the Formula, and How Total Momentum Stays Constant in Collisions

The law of conservation of momentum states that the total momentum of a closed system stays constant, if no net external force acts on the system. Total momentum before a collision equals total momentum after the collision.

For a reliable reference, see OpenStax explains when total linear momentum is conserved in a system.

Momentum calculations also use the reference equations in Physics 1 Formula Sheet: Equations, Constants, and Exam Rules for 8 Units.

What Is the Law of Conservation of Momentum?

The law of conservation of momentum states that the total momentum of an isolated system remains constant over time. Momentum is transferred between objects inside the system, and the total does not change.

A closed system exchanges no mass with its surroundings. An isolated system also experiences zero net external force. Examples of systems include 2 colliding carts, a rifle with its bullet, and a rocket with its exhaust gas.

What Is Momentum?

Momentum is the product of an object’s mass and velocity (p = mv). Momentum is a vector quantity, and it points in the direction of velocity. The SI unit of momentum is kg·m/s.

The table below lists the momentum of 4 objects. Rows list the objects, and columns show mass, velocity, and momentum.

Who Discovered the Law of Conservation of Momentum?

René Descartes proposed a conserved quantity of motion in 1644 in Principles of Philosophy. John Wallis, Christopher Wren, and Christiaan Huygens presented collision rules to the Royal Society in 1668 and 1669. Their rules accounted for direction.

Isaac Newton stated the law in 1687 in Principia, as a corollary to the laws of motion.

What Is the Formula for Conservation of Momentum?

The formula for conservation of momentum is p(initial) = p(final). For 2 objects, the formula expands to m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′.

The table below defines each symbol. Rows list the symbols, and columns show the quantity and the SI unit.

What Are the 3 Conditions for Conservation of Momentum?

Total momentum is conserved if the system is closed, the net external force is zero, and every interacting object is included.

1. Closed system: no mass enters or leaves.
2. Zero net external force: forces such as friction, air resistance, and gravity cancel or are negligible during the interaction.
3. Complete system: the calculation includes every object that exerts a force within the system.

Collisions last milliseconds, so external forces produce negligible impulse during the collision.

Does Conservation of Momentum Apply Along Each Axis?

Yes, momentum is conserved separately along each axis, if the net external force along that axis equals zero. A thrown ball keeps constant horizontal momentum, if air resistance is negligible. Its vertical momentum changes because gravity acts downward.

How Is Conservation of Momentum Derived From Newton’s Laws?

Conservation of momentum derives from Newton’s second and third laws. The derivation has 6 steps:

1. Consider 2 objects that exert forces only on each other.
2. Apply Newton’s third law to get F₁₂ = −F₂₁.
3. Write each force as the rate of change of momentum, F = dp/dt.
4. Substitute to get dp₁/dt = −dp₂/dt.
5. Add the terms to get d(p₁ + p₂)/dt = 0.
6. Conclude that p₁ + p₂ equals a constant.

How Does Impulse Relate to Conservation of Momentum?

Impulse equals the change in momentum of an object (J = FΔt = Δp). Each object in a collision receives an equal and opposite impulse. The momentum gained by one object equals the momentum lost by the other.

Airbags use this relation. A 70 kg occupant at 15 m/s has a momentum of 1,050 kg·m/s. Stopping in 0.1 s requires an average force of 10,500 N, and stopping in 0.2 s requires 5,250 N.

What Is the Connection Between Conservation of Momentum and Noether’s Theorem?

Noether’s theorem states that every continuous symmetry of a physical system corresponds to a conserved quantity. Emmy Noether proved the theorem in 1918. Conservation of linear momentum corresponds to translation symmetry in space, which means the laws of physics are identical at every location.

The table below lists 3 symmetries. Rows list each symmetry, and columns show the conserved quantity.

What Are the Types of Collisions in Conservation of Momentum?

There are 3 types of collisions: elastic, inelastic, and perfectly inelastic. Momentum is conserved in all 3 types, if the system is isolated.

The coefficient of restitution (e) equals the relative speed of separation divided by the relative speed of approach: e = (v₂′ − v₁′)/(v₁ − v₂).

The table below compares the 3 types. Rows list the collision types, and columns show momentum, kinetic energy, the state of the objects after impact, the coefficient of restitution, and examples.

What Is an Elastic Collision?

An elastic collision is a collision in which total momentum and total kinetic energy both stay constant. The 1D formulas for final velocities are:

• v₁′ = [(m₁ − m₂)/(m₁ + m₂)] v₁ + [2m₂/(m₁ + m₂)] v₂
• v₂′ = [2m₁/(m₁ + m₂)] v₁ + [(m₂ − m₁)/(m₁ + m₂)] v₂

Objects of equal mass exchange velocities in a head-on elastic collision.

What Is an Inelastic Collision?

An inelastic collision is a collision in which total momentum stays constant and total kinetic energy decreases. The lost kinetic energy converts to thermal energy, sound, and deformation. Car crashes are examples of inelastic collisions.

What Is a Perfectly Inelastic Collision?

A perfectly inelastic collision is a collision in which the objects stick together and move with a common final velocity. The formula is v = (m₁v₁ + m₂v₂)/(m₁ + m₂). Examples include a bullet embedded in a wooden block and 2 railway cars that couple.

How Do You Solve Conservation of Momentum Problems?

Solve conservation of momentum problems by equating total momentum before and after the interaction. The method has 6 steps:

1. Define the system, and confirm that the net external force equals zero.
2. Choose a positive direction for velocity.
3. Calculate the total momentum before the interaction.
4. Write the expression for total momentum after the interaction.
5. Solve for the unknown quantity.
6. Verify the direction and units, and compare kinetic energy before and after.

How Do You Solve a Perfectly Inelastic Collision Problem?

Example 1: A 1,000 kg car moving at 20 m/s collides with a stationary 1,500 kg car, and the cars lock together.

• Initial momentum: 1,000 × 20 + 1,500 × 0 = 20,000 kg·m/s
• Final velocity: v = 20,000 ÷ 2,500 = 8 m/s
• Initial kinetic energy: ½ × 1,000 × 20² = 200,000 J
• Final kinetic energy: ½ × 2,500 × 8² = 80,000 J
• Kinetic energy lost: 120,000 J, which is 60% of the initial value

How Do You Solve an Elastic Collision Problem?

Example 2: A 2 kg object moving at 4 m/s collides head-on and elastically with a stationary 1 kg object.

• v₁′ = [(2 − 1)/(2 + 1)] × 4 = 1.33 m/s
• v₂′ = [(2 × 2)/(2 + 1)] × 4 = 5.33 m/s
• Momentum check: 2 × 1.33 + 1 × 5.33 = 8 kg·m/s, which equals the initial 2 × 4 = 8 kg·m/s
• Kinetic energy check: initial 16 J, and final 1.78 J + 14.22 J = 16 J

How Do You Solve a Recoil Problem?

Example 3: A 4 kg rifle at rest fires a 0.02 kg bullet at 500 m/s.

• Initial momentum: 0
• Equation: 0 = 0.02 × 500 + 4 × v
• Recoil velocity: v = −10 ÷ 4 = −2.5 m/s

The rifle recoils at 2.5 m/s in the direction opposite to the bullet.

How Do You Solve a Two-Dimensional Collision Problem?

Example 4: A 2 kg object moving east at 3 m/s collides with a 1 kg object moving north at 6 m/s, and the objects stick together.

• East component: 2 × 3 = 3 × vₓ, so vₓ = 2 m/s
• North component: 1 × 6 = 3 × vᵧ, so vᵧ = 2 m/s
• Final speed: √(2² + 2²) = 2.83 m/s, at 45° north of east

What Is Conservation of Momentum in an Explosion?

In an explosion, internal forces separate the fragments, and the total momentum of all fragments equals the momentum before the explosion. An object at rest has zero initial momentum, so the fragments carry equal and opposite momenta.

Example 5: A 6 kg object at rest breaks into a 4 kg fragment and a 2 kg fragment. The 4 kg fragment moves east at 3 m/s.

• Equation: 0 = 4 × 3 + 2 × v
• Velocity of the 2 kg fragment: v = −6 m/s, which is 6 m/s west
• Kinetic energy after the explosion: ½ × 4 × 3² + ½ × 2 × 6² = 18 J + 36 J = 54 J

Kinetic energy increases from 0 J to 54 J, and the source is stored chemical or elastic potential energy. Total energy stays constant.

What Are Examples of Conservation of Momentum in Daily Life?

Conservation of momentum appears whenever objects interact with negligible external force. There are 6 common examples:

1. Billiard balls: the cue ball transfers momentum to the object ball on impact.
2. Rocket launches: downward exhaust gas momentum equals upward rocket momentum.
3. Firearm recoil: forward bullet momentum equals backward firearm momentum.
4. Car collisions: total momentum of both cars is equal before and after impact.
5. Ice skaters pushing apart: a 60 kg skater moving at 2 m/s and a 40 kg skater moving at 3 m/s in the opposite direction each have 120 kg·m/s of momentum.
6. Newton’s cradle: a swinging ball passes momentum through the stationary balls to the end ball.

How Does Conservation of Momentum Explain Rocket Propulsion?

A rocket gains forward momentum equal to the backward momentum of its exhaust gas. The system consists of the rocket and the exhaust, so total momentum stays constant.

Konstantin Tsiolkovsky derived the rocket equation in 1903: Δv = vₑ ln(m₀/m_f). The symbol vₑ is exhaust velocity, m₀ is initial mass, and m_f is final mass. A rocket with vₑ = 3,000 m/s and a mass ratio of 5 gains Δv = 3,000 × ln(5) = 4,828 m/s.

NASA Glenn Research Center explains rocket thrust through Newton’s laws of motion.

Where Is Conservation of Momentum Used in Science and Engineering?

Conservation of momentum is used to calculate unknown velocities, forces, and masses. There are 5 main applications:

1. Accident reconstruction: investigators estimate pre-crash speeds from vehicle masses and post-crash motion.
2. Ballistics: the ballistic pendulum, introduced by Benjamin Robins in 1742, measures bullet speed from the swing of a block.
3. Spacecraft maneuvers: thrusters change velocity by expelling mass.
4. Particle physics: detectors infer neutrinos from missing transverse momentum. Wolfgang Pauli proposed the neutrino in 1930 to account for apparently missing energy in beta decay, and Clyde Cowan and Frederick Reines detected it in 1956.
5. Sports engineering: bat-ball and club-ball collisions determine launch speed.

What Is the Difference Between Conservation of Momentum and Conservation of Energy?

Conservation of momentum applies to a vector quantity in systems with zero net external force. Conservation of energy applies to a scalar quantity in systems with no energy transfer across the boundary.

The table below compares the 2 laws. Rows list the properties, and columns show the values for momentum and energy.

What Is the Difference Between Linear and Angular Momentum Conservation?

Linear momentum conservation applies to translation (p = mv), and angular momentum conservation applies to rotation (L = Iω). The symbol I is moment of inertia, and ω is angular velocity.

Angular momentum stays constant if the net external torque equals zero. A skater who reduces moment of inertia from 4 kg·m² to 2 kg·m² doubles angular speed from 3 rad/s to 6 rad/s, because 4 × 3 = 2 × 6 = 12 kg·m²/s.

How Does Conservation of Momentum Work in Relativity and Quantum Mechanics?

Conservation of momentum holds in special relativity and quantum mechanics, with modified formulas. Relativistic momentum is p = γmv, where γ = 1/√(1 − v²/c²).

Photons have momentum p = h/λ, where h is the Planck constant and λ is wavelength. Arthur Compton confirmed photon momentum in 1923 by measuring X-ray scattering from electrons. Total momentum of the photon and electron stays constant in each scattering event.

What Are the Limitations of the Law of Conservation of Momentum?

The law applies only to systems with zero net external force along the axis considered. There are 4 main limitations:

1. External forces: friction, gravity, and applied pushes change total momentum.
2. Incomplete systems: omitting an interacting object produces an apparent violation. A falling ball gains downward momentum, and Earth gains equal upward momentum.
3. Long interaction times: external impulse becomes non-negligible as duration increases.
4. Non-inertial frames: fictitious forces alter momentum balance.

What Are Common Misconceptions About Conservation of Momentum?

Physics education research documents persistent misconceptions about collisions. The Force Concept Inventory, published in 1992 by Hestenes, Wells, and Swackhamer, includes questions about collision forces. Many students select the answer that the heavier object exerts a larger force.

The table below lists 4 misconceptions. Rows show each misconception, and columns show the incorrect belief and the correct fact.

People Also Ask About Conservation of Momentum

Is Momentum Conserved in an Inelastic Collision?

Yes, momentum is conserved in all collisions in an isolated system, while kinetic energy decreases in inelastic collisions.

What Is the SI Unit of Momentum?

The SI unit of momentum is kg·m/s, which equals N·s.

Is Momentum a Vector or a Scalar Quantity?

Momentum is a vector quantity with magnitude and direction.

Why Is Momentum Conserved?

Momentum is conserved because internal forces between objects are equal and opposite by Newton’s third law, so they cancel in the total.

Is Momentum Conserved When Kinetic Energy Is Lost?

Yes, total momentum stays constant when kinetic energy converts to other forms of energy.

What Is the Formula for Conservation of Momentum?

The formula is m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′ for 2 interacting objects.

What Is the Main Summary of Conservation of Momentum?

The law of conservation of momentum states that total momentum of an isolated system stays constant. There are 6 key facts:

1. Momentum equals mass times velocity, and the SI unit is kg·m/s.
2. Total momentum is conserved in elastic, inelastic, and perfectly inelastic collisions.
3. Kinetic energy is conserved only in elastic collisions.
4. Momentum is conserved along each axis separately, if the net external force along that axis equals zero.
5. Newton’s third law and Noether’s theorem explain the law.
6. Rockets, explosions, and recoil follow the same law.

Sources:

• René Descartes, Principles of Philosophy (1644)
• Isaac Newton, Philosophiæ Naturalis Principia Mathematica (1687)
• Wallis, Wren, and Huygens, collision rules, Royal Society, Philosophical Transactions (1668 and 1669)
• Benjamin Robins, New Principles of Gunnery (1742)
• Konstantin Tsiolkovsky, "Exploration of Outer Space by Means of Rocket Devices" (1903)
• Emmy Noether, "Invariante Variationsprobleme" (1918)
• Arthur Compton, Physical Review (1923)
• Cowan, Reines, et al., "Detection of the Free Neutrino," Science (1956)
• Hestenes, Wells, and Swackhamer, "Force Concept Inventory," The Physics Teacher (1992)
• NASA Glenn Research Center, rocket thrust educational pages
• Bureau International des Poids et Mesures (BIPM), SI Brochure

Maurice Cotterell

MEd., BSc. QTS/OCT – Physics Teacher at Davenant Foundation School
The University of the West Indies

My passion as a teacher and leader in education is to contribute to students’ enhanced learning by creation of a positive teaching/learning environment, where they are motivated to develop strong critical thinking, analytical and problem-solving skills.
As acting Vice-Principal and head of Physics, I have amassed five years of experience in leading meetings, implementing strategies to maintain discipline and school ethos, liaise with stakeholders in planning for future development of the institution; successfully negotiated the implementation of various projects for student development as well as supervising effective execution of such. For the past 19 years, I have incorporated discovery learning within the classroom through practical activities, problem solving and cooperative learning and have realized that students retention in Physics and Mathematics have increased significantly.

I am always cognizant that students have different learning styles, and as such teaching strategies during planning and delivery must take this crucial factor into consideration and be tailored to meet such differences.