Conservation of Momentum Calculator
Solve the 1D elastic collision relation m₁·v₁ + m₂·v₂ = m₁·v₁′ + m₂·v₂′ for any of its six unknowns — two masses, two initial velocities, and two final velocities — with a live collision schematic, step-by-step math, charts, and companion mechanics metrics.
Momentum is conserved in every collision
The total momentum of an isolated system is conserved in every collision — elastic, inelastic, or explosive — as long as no external net force acts on the system. This is a direct consequence of Newton's third law: the forces the two bodies exert on each other during the collision are equal and opposite, so the total change in momentum is zero. Momentum conservation is more general than energy conservation, because it holds even when kinetic energy is lost to heat, sound, or deformation. In this calculator we focus on elastic collisions, where kinetic energy is also conserved.
Elastic collisions conserve kinetic energy too
An elastic collision is one in which both momentum and kinetic energy are conserved. This requires that no energy is lost to deformation, heat, or sound — an idealisation that is approached by billiard balls, gas molecules, and subatomic particles, but never perfectly achieved by macroscopic objects. For a 1D elastic collision, the two conservation laws (momentum and energy) give two equations, which can be solved simultaneously to determine the two final velocities from the initial masses and velocities. The result is the pair of formulas in the "How to use" section: v₁′ = ((m₁−m₂)v₁ + 2m₂v₂)/(m₁+m₂) and v₂′ = ((m₂−m₁)v₂ + 2m₁v₁)/(m₁+m₂).
Equal masses exchange velocities
When m₁ = m₂, the elastic formulas simplify dramatically: v₁′ = v₂ and v₂′ = v₁. The two bodies simply exchange velocities. If the target is stationary (v₂ = 0), the incident mass stops dead and the target moves off with the incident velocity — the classic "billiard ball" collision. This is the most familiar case in demonstrations and in games of pool or snooker. It is a special property of equal masses in 1D elastic collisions, and it also holds in the center-of-mass frame for the components perpendicular to the collision axis in 2D and 3D elastic collisions.
Center-of-mass frame simplifies everything
In the center-of-mass (CM) frame — the reference frame that moves with the system's center of mass — the total momentum is zero by definition, and the two masses always approach each other and recede with equal and opposite momenta. In this frame, an elastic collision simply reverses the direction of each particle's velocity without changing its magnitude. The CM velocity itself is v_cm = (m₁v₁ + m₂v₂)/(m₁ + m₂), and it is unchanged by the collision (a consequence of momentum conservation). Working in the CM frame often simplifies the algebra of collision problems and reveals the underlying symmetry of the interaction.
How to Use This Calculator
The conservation of momentum relation links six quantities: m₁, m₂, v₁, v₂, v₁′, and v₂′. Pick which one you want the tool to solve, then fill in the other five.
1. Choose what to solve for
Pick any of the six quantities from the toggle. The calculator rearranges m₁·v₁ + m₂·v₂ = m₁·v₁′ + m₂·v₂′ automatically for whichever you choose. When solving for a final velocity, the elastic-collision energy condition is used as an additional equation.
2. Use realistic presets
Presets cover typical values: masses from 0.1 to 10 kg, initial velocities from −5 to +20 m/s, and final velocities from −4 to +8 m/s. Use the ones closest to your scenario and adjust.
3. Watch the sign convention
Velocities are signed: positive means moving to the right, negative to the left. The direction matters, and a negative final velocity is perfectly physical (a bounce-back). The calculator preserves signs throughout and never takes absolute values, so momentum conservation is respected exactly.
4. Read the result and verdict
The result card shows the solved value plus a colour-coded verdict that checks whether the inputs are consistent with an elastic collision. If the initial state has enough kinetic energy for the required final state, the verdict is green; otherwise, a warning is shown.
5. Check the schematic and charts
The schematic shows the "before" and "after" states of the two masses with their velocity arrows, plus the total momentum and kinetic energy. The chart compares the initial and final velocities of both bodies and the center-of-mass velocity.
6. Switch to Advanced for depth
Advanced mode adds a chart of final velocities vs. mass ratio, a sensitivity check on a +1% input change, and companion metrics including kinetic energy before and after, the Q-value (energy loss), the CM-frame velocities, and the impulse on each mass.
What This Tool Offers
Built around the conservation of momentum relation m₁·v₁ + m₂·v₂ = m₁·v₁′ + m₂·v₂′, the fundamental link between initial and final states in a 1D collision.
Six-way solving
Solve for either mass, either initial velocity, or either final velocity. The calculator rearranges the conservation law for whichever variable you choose.
Clean collision schematic
"Before" and "after" states of the two masses, with velocity arrows, total momentum, total kinetic energy, and the center-of-mass velocity.
Consistency verdict
Each result is checked against the elastic-collision energy condition and given a colour-coded verdict: valid, energy-deficient, or non-physical.
Multi-unit hints
Each field shows the value in related forms — kg for masses, m/s and km/h for velocities, and the corresponding momentum and kinetic-energy contributions.
Dual comparison charts
A before/after velocity comparison, plus an advanced chart of final velocities vs. mass ratio for the general elastic case.
Reference tables
Elastic collision results for special cases (equal masses, stationary target, heavy target), and a full variable reference table.
Reference Tables & Conversions
A momentum unit converter, elastic collision lookup grids, and special cases.
Momentum unit converter
Elastic Collision Results at m₁ = 2 kg, v₁ = 4 m/s, v₂ = −1 m/s
Computed from the momentum and energy conservation laws. Values in m/s.
| m₂ (kg) | v₁′ (m/s) | v₂′ (m/s) | p_total before | p_total after | KE before (J) | KE after (J) |
|---|
Special Cases of 1D Elastic Collisions (stationary target, v₂ = 0)
| Case | v₁′ | v₂′ | Notes |
|---|---|---|---|
| Equal masses (m₁ = m₂) | 0 | v₁ | Incident mass stops; target moves off with v₁ |
| Very light projectile (m₁ ≪ m₂) | −v₁ | ≈ 0 | Projectile bounces back at nearly the same speed |
| Very heavy projectile (m₁ ≫ m₂) | ≈ v₁ | ≈ 2v₁ | Projectile barely slows; target flies off at ≈ 2v₁ |
| General elastic (v₂ = 0) | (m₁−m₂)v₁ / (m₁+m₂) | 2m₁v₁ / (m₁+m₂) | Standard elastic collision formula |
| Perfectly inelastic (v₂ = 0) | m₁v₁ / (m₁+m₂) | m₁v₁ / (m₁+m₂) | Both masses move together; KE lost to heat |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| m₁ | Mass of body 1 | 10⁻³ – 10⁴ | kg |
| m₂ | Mass of body 2 | 10⁻³ – 10⁴ | kg |
| v₁ | Initial velocity of mass 1 | −10³ – 10³ | m/s |
| v₂ | Initial velocity of mass 2 | −10³ – 10³ | m/s |
| v₁′ | Final velocity of mass 1 | −10³ – 10³ | m/s |
| v₂′ | Final velocity of mass 2 | −10³ – 10³ | m/s |
| p | Total momentum | — | kg·m/s |
| KE | Total kinetic energy | — | J |
| vcm | Center-of-mass velocity | — | m/s |
What Does m₁·v₁ + m₂·v₂ = m₁·v₁′ + m₂·v₂′ Measure?
This equation is the statement of conservation of momentum for a two-body collision in one dimension. It says that the total momentum of the system — the vector sum of the momenta of the two bodies — is the same before and after the collision. This conservation law is one of the most fundamental in physics, holding in every collision regardless of the forces involved, as long as no external net force acts on the system. For an elastic collision, it is supplemented by the conservation of kinetic energy, and the two laws together determine the final velocities uniquely.
Conservation of Momentum Formula
The defining relationship for a 1D two-body collision is:
For an elastic collision, kinetic energy is also conserved:
Solving these two equations simultaneously gives the final velocities:
v₂′ = ( ( m₂ − m₁ ) v₂ + 2 m₁ v₁ ) / ( m₁ + m₂ )
The center-of-mass velocity and the total momentum are:
p = m₁ v₁ + m₂ v₂ = ( m₁ + m₂ ) vcm
In the center-of-mass frame, the relative velocity simply reverses:
Collision Variables
| Symbol | Meaning | Notes |
|---|---|---|
| m₁, m₂ | Masses of the two bodies | Always positive; conserved in classical mechanics |
| v₁, v₂ | Initial velocities | Signed; positive to the right, negative to the left |
| v₁′, v₂′ | Final velocities | Solved from conservation laws |
| p | Total momentum | Conserved in every collision (isolated system) |
| KE | Total kinetic energy | Conserved only in elastic collisions |
| vcm | Center-of-mass velocity | Unchanged by the collision |
| Q | Energy loss (Q-value) | Q = KE_initial − KE_final; zero for elastic |
How Each Variable Affects the Collision
Momentum conservation alone does not determine the final state — an infinite number of (v₁′, v₂′) pairs satisfy it, corresponding to different energy losses. The nature of the collision (elastic, partially inelastic, or perfectly inelastic) selects one of these pairs. For an elastic collision, kinetic energy conservation picks out a unique pair, and the final velocities can be written explicitly in terms of the masses and initial velocities. The key ratios are m₂/m₁ and (v₂ − v₁): the first determines how much momentum each mass carries, and the second is the relative approach speed that governs the energy available for the collision.
The mass ratio has a particularly intuitive interpretation. When m₁ = m₂, the two bodies exchange velocities in an elastic collision. When m₁ < m₂ and the target is at rest, the incident mass bounces back (v₁′ is negative), and in the limit m₁ ≪ m₂ it bounces back at nearly its original speed. When m₁ > m₂ and the target is at rest, the incident mass continues forward (v₁′ is positive), and in the limit m₁ ≫ m₂ it barely slows while the target flies off at roughly twice the incident speed. These limits are the basis of many practical rules of thumb in billiards, ballistics, and particle physics.
Conservation of Momentum Calculation Examples
v₁′ = ((2−3)(4) + 2·3·(−1))/(2+3) = (−4 − 6)/5 = −2 m/s
v₂′ = ((3−2)(−1) + 2·2·(4))/(2+3) = (−1 + 16)/5 = 3 m/s
v₁′ = 0 m/s, v₂′ = 5 m/s (velocities exchanged)
v₁′ ≈ −99 m/s (bounces back), v₂′ ≈ 0.2 m/s
v₁′ ≈ 4.99 m/s (barely slows), v₂′ ≈ 9.99 m/s
Frequently Asked Questions
Momentum is a vector, p = mv, with units of kg·m/s (or N·s). Kinetic energy is a scalar, KE = ½mv², with units of joules. Momentum is always conserved in an isolated system; kinetic energy is only conserved in elastic collisions. During an inelastic collision, momentum is transferred between the bodies but the total is unchanged, while kinetic energy is converted into heat, sound, or deformation. This is why momentum conservation can be used to solve for the final state even when energy is lost, but energy conservation alone cannot determine the outcome of a collision without additional information about the collision's nature.
An elastic collision is one in which the total kinetic energy of the system is the same before and after the collision. This requires that no energy is converted into heat, sound, deformation, or other forms. Perfect elastic collisions do not occur for macroscopic objects, but they are closely approached by collisions between hard, smooth spheres (like billiard balls), by collisions between gas molecules (which is why the kinetic theory of gases assumes elastic collisions), and by collisions between subatomic particles. For elastic collisions, the two conservation laws (momentum and energy) give two equations that uniquely determine the two final velocities.
In the center-of-mass frame, an elastic collision simply reverses the direction of each particle's velocity without changing its magnitude — a consequence of the symmetry of the elastic interaction. This means the relative velocity (the difference between the two velocities) has the same magnitude before and after, but opposite sign. In the lab frame, the same statement becomes v₁′ − v₂′ = −(v₁ − v₂). This "reversal of relative velocity" is a compact way to state the combined effect of momentum and energy conservation in an elastic collision, and it is often more convenient to use than the full algebraic formulas for v₁′ and v₂′.
The center-of-mass (CM) frame is the reference frame in which the total momentum of the system is zero. It moves with velocity v_cm = (m₁v₁ + m₂v₂)/(m₁ + m₂) relative to the lab frame. In the CM frame, the two bodies always approach each other with equal and opposite momenta, and in an elastic collision they simply reverse direction without changing speed. This makes the physics of the collision much simpler and reveals its underlying symmetry. Many problems in collision physics — including scattering angles, energy transfer, and inelastic thresholds — are most easily solved in the CM frame and then transformed back to the lab frame.
Yes, as long as the system is isolated — that is, no net external force acts on it. The internal forces during the collision (the forces the two bodies exert on each other) are equal and opposite by Newton's third law, so they always cancel in the total momentum balance. This holds for elastic, inelastic, and explosive processes alike. Only when an external force is present (friction with the ground, air resistance, gravity during a long collision) does the total momentum change, and then the change equals the impulse of the external force. For collisions that are brief compared to the timescale of external forces, the external impulse is negligible and momentum conservation is an excellent approximation.
A perfectly inelastic collision is one in which the two bodies stick together and move with a common final velocity after the collision. This is the maximum possible loss of kinetic energy consistent with momentum conservation. The common velocity is v′ = (m₁v₁ + m₂v₂)/(m₁ + m₂) = v_cm, and the kinetic energy lost is ΔKE = ½ (m₁m₂/(m₁+m₂)) (v₁ − v₂)². This lost energy goes into deformation, heat, and sound. Perfectly inelastic collisions are the opposite extreme from elastic collisions, and real collisions lie somewhere in between, characterised by a coefficient of restitution e between 0 (perfectly inelastic) and 1 (perfectly elastic).
In two or three dimensions, momentum conservation becomes a vector equation: the total momentum vector is conserved component by component, giving two or three scalar equations. For an elastic collision, energy conservation adds one more equation, but the number of unknowns (the final velocity vectors of the two bodies) is four or six, so the problem is underdetermined without additional information — such as the direction of one of the final velocities or the geometry of the impact. This is why 2D and 3D collision problems typically require more information than 1D problems. The 1D case is special because the number of equations (2) matches the number of unknowns (2).
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| m1, m2 | Masses of the two colliding objects | kg |
| v1, v2 | Velocities of the objects before collision | m/s |
| v1', v2' | Velocities of the objects after collision | m/s |
What it means
In an elastic collision, both momentum and kinetic energy are conserved. For a one‑dimensional collision between two objects, the sum of their momenta before collision equals the sum after. This equation, along with the conservation of kinetic energy (½m₁v₁² + ½m₂v₂² = ½m₁v₁'² + ½m₂v₂'²), allows solving for the final velocities. Elastic collisions occur between macroscopic objects like billiard balls or between atoms/molecules. The equations are used in physics to analyse scattering processes, and in engineering to design impact‑absorbing systems. They also apply in particle physics to study fundamental interactions. Understanding conservation laws is fundamental to mechanics and many other branches of physics.
Worked example
Conservation of Momentum – Two Examples
Real‑World| Parameter | Value |
|---|---|
| m₁ | 2 kg |
| v₁ | 5 m/s |
| m₂ | 3 kg |
| v₂ | -2 m/s |
| Parameter | Value |
|---|---|
| m₁ | 1 kg |
| v₁ | 10 m/s |
| m₂ | 1 kg |
| v₂ | 0 m/s |
Common mistakes
- Elastic collision: This equation only states momentum conservation – for a 1D elastic collision, energy is also conserved (additional equation).
- Velocities: v₁, v₂ are initial; v₁ʹ, v₂ʹ are final – use consistent sign conventions (positive direction).
- Masses: m₁ and m₂ in kg.
- Perfectly elastic: If not perfectly elastic, the equation still holds for momentum, but energy is not conserved – use coefficient of restitution.
- Vector: In 1D, use scalar with signs; in 2D, treat components separately.
Applications
Conservation of momentum in a one‑dimensional elastic collision, m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂', states that total momentum is conserved when no external forces act. This principle is essential for analysing collisions in particle physics, vehicle impacts, and sports. Engineers use it to design safety systems that manage momentum transfer during crashes. In atomic and nuclear physics, it helps interpret scattering experiments. In sports, it explains the behaviour of billiard balls and the transfer of energy in tennis. The equation is also applied in the design of propulsion systems where momentum exchange is key. By using this conservation law, professionals can predict post‑collision velocities and design systems that achieve desired outcomes while respecting physical constraints.
- Vehicle collision reconstruction and safety analysis
- Design of bumper systems and crash barriers
- Particle scattering experiments in physics
- Sports equipment design (billiards, tennis, golf)
- Rocket staging and multi‑body dynamics
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