Linear Momentum Calculator
Solve the linear momentum relation p = m · v for any of its three unknowns — momentum, mass, or velocity — with step-by-step math, a live vector schematic, charts, and companion mechanics metrics.
Momentum is the "quantity of motion"
Linear momentum is the product of an object's mass and its velocity, p = mv. It was called the "quantity of motion" by Newton and is the natural measure of how hard it is to stop a moving object. A heavy truck at low speed and a light bullet at high speed can have the same momentum, and they require the same impulse to stop. Momentum is a vector: its direction is the direction of the velocity, and its sign tells you which way the object is moving.
Momentum is conserved in isolated systems
In the absence of external forces, the total momentum of a system is conserved. This is one of the most fundamental laws of physics, and it holds in every collision, explosion, and interaction — elastic or inelastic, classical or relativistic. The conservation of momentum is a consequence of Newton's third law: the internal forces between parts of a system are equal and opposite, so their total impulse is zero. This is why rockets work: the momentum of the exhaust gases backward equals the momentum of the rocket forward.
Momentum vs. kinetic energy
Momentum p = mv and kinetic energy KE = ½mv² are both measures of motion, but they behave differently. Momentum is linear in velocity, kinetic energy is quadratic. Doubling the speed doubles the momentum but quadruples the kinetic energy. Momentum is a vector, kinetic energy is a scalar. Both are conserved in elastic collisions, but only momentum is conserved in inelastic collisions — kinetic energy can be converted into heat, sound, or deformation. This is why a bullet has much more momentum than kinetic energy relative to a slow, heavy object of the same kinetic energy.
Relativistic momentum at high speeds
At speeds close to the speed of light, the classical formula p = mv breaks down. The correct relativistic expression is p = γmv, where γ = 1/√(1 − v²/c²) is the Lorentz factor. As v approaches c, γ grows without bound, so the momentum also grows without bound — this is why no massive object can reach the speed of light. For everyday speeds (v < 0.01c ≈ 3×10⁶ m/s), the factor γ differs from 1 by less than 0.005%, and the classical formula is an excellent approximation. In particle accelerators, however, relativistic momentum is essential and is routinely measured in units of GeV/c.
How to Use This Calculator
The linear momentum relation links three quantities: p, m, and v. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick p, m, or v from the toggle. The calculator rearranges p = m·v automatically for whichever you choose.
2. Use realistic presets
Presets cover typical values: momentum from 0.1 to 10⁵ kg·m/s, mass from 1 g to 1,500 kg, and velocity from −10 to +100 m/s. Use the ones closest to your scenario and adjust.
3. Watch the sign convention
Momentum and velocity are signed vectors: positive means moving to the right, negative to the left. The direction matters for conservation laws and collision problems. The calculator preserves signs throughout, so p = m·v is satisfied with the correct sign.
4. Read the result and verdict
The result card shows the solved value plus equivalent units. A colour-coded verdict classifies the momentum from very small (insect-scale) through moderate (everyday objects) to very large (vehicle and rocket scale).
5. Check the schematic and charts
The schematic shows the object with its mass and velocity vectors, and the resulting momentum vector scaled in proportion. The chart plots momentum against velocity for three values of mass.
6. Switch to Advanced for depth
Advanced mode adds a chart of p vs. m, a sensitivity check on a +1% input change, and companion metrics including kinetic energy, the momentum-to-energy ratio, de Broglie wavelength, and relativistic momentum for comparison.
What This Tool Offers
Built around the linear momentum relation p = m · v, the fundamental link between mass, velocity, and the quantity of motion.
Three-way solving
Solve for momentum, mass, or velocity. The calculator rearranges the formula for whichever variable you choose.
Clean vector schematic
The object with mass m and velocity vector v, and the resulting momentum vector p, all scaled in proportion.
Momentum-class verdict
Each result is classified from very small (insect-scale) through moderate (everyday) to very large (vehicle and rocket scale), with a coloured badge.
Multi-unit hints
Each field shows the value in related forms — N·s and lb·ft/s for momentum, g and lb for mass, km/h and mph for velocity.
Dual comparison charts
p vs. v for three values of m, plus an advanced p vs. m chart for three values of v.
Reference tables
Typical momentum values by scenario, a momentum unit converter, and a full variable reference table.
Reference Tables & Conversions
A momentum unit converter, a lookup grid, and typical momentum values.
Momentum unit converter
Momentum Lookup at m = 5 kg
Computed from p = m · v. Values in kg·m/s.
| v (m/s) | m = 0.001 kg | m = 0.1 kg | m = 1 kg | m = 5 kg | m = 100 kg |
|---|
Typical Momentum Values by Scenario
| Object | Mass (kg) | Velocity (m/s) | Momentum (kg·m/s) | Notes |
|---|---|---|---|---|
| Flying mosquito | 2×10⁻⁶ | 1 | 2×10⁻⁶ | Extremely small |
| Bullet (9 mm) | 0.008 | 350 | 2.8 | Small but fast |
| Tennis ball serve | 0.058 | 60 | 3.5 | Fast sports object |
| Soccer ball kick | 0.43 | 30 | 12.9 | Common sports |
| Human walk | 70 | 1.4 | 98 | Everyday scale |
| Human sprint | 70 | 10 | 700 | Elite athlete |
| Car (50 km/h) | 1,500 | 13.9 | 20,850 | Everyday vehicle |
| Truck (80 km/h) | 30,000 | 22.2 | 666,000 | Heavy vehicle |
| Boeing 747 at cruise | 300,000 | 250 | 7.5×10⁷ | Large aircraft |
| Saturn V rocket | 2.9×10⁶ | 2,400 | 7×10⁹ | Launch vehicle |
| Earth in orbit | 6×10²⁴ | 30,000 | 1.8×10²⁹ | Astronomical |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| p | Linear momentum | 10⁻⁶ – 10²⁹ | kg·m/s = N·s |
| m | Mass | 10⁻³⁰ – 10²⁴ | kg |
| v | Velocity | 0 – 3×10⁸ | m/s |
| KE | Kinetic energy = ½mv² | — | J |
| p²/2m | Kinetic energy in terms of p | — | J |
| γ | Lorentz factor = 1/√(1−v²/c²) | — | — |
| λ | de Broglie wavelength = h/p | — | m |
What Does p = m · v Measure?
Linear momentum is the product of an object's mass and its velocity. It is the natural measure of "how much motion" an object has, and it plays a central role in mechanics because it is conserved in isolated systems. Momentum is a vector, so its direction matters, and it combines with the conservation laws for energy and angular momentum to give a complete description of motion in classical mechanics.
Linear Momentum Formula
The defining relationship is:
Rearranged to solve for the other two unknowns:
v = p / m
Momentum is related to kinetic energy and to Newton's second law:
F = dp / dt
J = Δp = ∫ F dt
In a system of several particles, the total momentum is the vector sum of the individual momenta:
At relativistic speeds, the momentum is:
Linear Momentum Variables
| Symbol | Meaning | Notes |
|---|---|---|
| p | Linear momentum | Vector; direction of velocity; kg·m/s = N·s |
| m | Mass | Scalar; always positive; kg |
| v | Velocity | Vector; signed; m/s |
| KE | Kinetic energy | Scalar; ½mv² = p²/(2m); J |
| F | Force | Rate of change of momentum; N |
| J | Impulse | Change in momentum; J = Δp; N·s |
| γ | Lorentz factor | Relativistic correction; γ ≥ 1 |
How Each Variable Affects Momentum
Momentum is directly proportional to both mass and velocity. Doubling the mass doubles the momentum; doubling the velocity also doubles the momentum. This linearity is what distinguishes momentum from kinetic energy, which is proportional to the square of velocity. A truck moving at 5 m/s has the same momentum as a car of half its mass moving at 10 m/s, but their kinetic energies are different: the car has twice the kinetic energy of the truck. This is why momentum and kinetic energy are both needed to fully characterise a moving object, and why both conservation laws are used together in collision analysis.
Because momentum is a vector, its direction matters. Two objects with equal mass and opposite velocities have equal and opposite momenta; their total momentum is zero. In an isolated system, this total momentum is conserved: if two objects collide and one speeds up, the other must slow down or reverse direction such that the vector sum remains unchanged. This is the essence of the conservation of momentum, and it is what makes momentum such a powerful tool for analysing collisions, explosions, and rocket propulsion. The momentum of the rocket exhaust is equal and opposite to the momentum gained by the rocket, so the total momentum of the rocket-plus-fuel system remains constant even as the rocket accelerates.
Linear Momentum Calculation Examples
p = 5 × 4 = 20 kg·m/s
m = 20 / 4 = 5 kg
v = 20 / 5 = 4 m/s
p = 0.008 × 350 = 2.8 kg·m/s
p = 1500 × 13.9 = 20,850 kg·m/s
p_total = 5×4 + 5×(−4) = 0 kg·m/s
Frequently Asked Questions
Velocity is a kinematic quantity: it describes how fast and in what direction an object is moving. Momentum is a dynamic quantity: it describes how much "motion" the object has, taking into account both its velocity and its mass. Two objects can have the same velocity but very different momenta if their masses differ — a truck and a bicycle both moving at 10 m/s have vastly different momenta. Conversely, two objects can have the same momentum with very different velocities if their masses compensate. Momentum is what determines how much force is needed to stop an object over a given time, and it is what is conserved in collisions.
Momentum conservation follows directly from Newton's third law: when two objects interact, the force one exerts on the other is equal in magnitude and opposite in direction. Because the forces act for the same duration, the impulses they deliver are equal and opposite, so the changes in momentum are equal and opposite, and the total momentum is unchanged. This argument generalises to any number of objects and to any internal forces, so the total momentum of an isolated system (one with no net external force) is always conserved. This is one of the most robust laws in physics, holding in elastic and inelastic collisions, in explosions, in rocket propulsion, and even in relativistic and quantum systems.
For a single object, the kinetic energy can be expressed in terms of momentum: KE = p²/(2m). This shows that for a fixed momentum, a lighter object has more kinetic energy than a heavier one. For example, a bullet and a truck can have the same momentum, but the bullet has much more kinetic energy because its mass is small. Conversely, for a fixed kinetic energy, momentum grows as the square root of mass: p = √(2m·KE). In a system of particles, total momentum and total kinetic energy are separate quantities: momentum is always conserved in an isolated system, but kinetic energy is only conserved in elastic collisions. In inelastic collisions, kinetic energy is lost but momentum is not.
Newton's second law can be written in its most general form as F = dp/dt: force equals the rate of change of momentum. This is more fundamental than the familiar F = ma, because it applies even when the mass is changing (as in a rocket burning fuel) and even at relativistic speeds. Integrating this over time gives the impulse-momentum theorem: ∫F dt = Δp. So a force acting over a period of time changes the momentum of an object, and the change equals the impulse delivered. This is why the momentum of an object can only change if a net external force acts on it, and why the total momentum of an isolated system (no net external force) is conserved.
Momentum is a vector, so in two or three dimensions it has components in each direction: p = (px, py, pz) = (mvx, mvy, mvz). The conservation of momentum applies separately to each component: the total x-momentum is conserved, the total y-momentum is conserved, and the total z-momentum is conserved, all independently. This component-wise conservation is what allows us to solve 2D and 3D collision problems by treating them as independent 1D problems in each direction. A classic example is a two-dimensional billiard collision: the momentum in the direction of the initial motion and the momentum perpendicular to it are both conserved, giving two equations that determine the final velocities.
Yes, significantly. At speeds approaching the speed of light, the classical formula p = mv must be replaced by the relativistic expression p = γmv, where γ = 1/√(1 − v²/c²) is the Lorentz factor. As v approaches c, γ grows without bound, so the momentum also grows without bound: no matter how much energy you pump into a particle, its speed approaches c but never reaches it, while its momentum continues to grow. This is why the momentum of a particle in a modern accelerator is quoted in units of GeV/c rather than kg·m/s — the numbers would be unwieldy otherwise. For everyday speeds (below about 1% of c), γ differs from 1 by less than 0.005%, and the classical formula is essentially exact.
In quantum mechanics, every particle has a wavelength associated with its momentum, given by the de Broglie relation λ = h/p, where h = 6.626×10⁻³⁴ J·s is Planck's constant. This is why electrons can diffract and interfere like waves, and it is the basis of electron microscopy and quantum mechanics. For a macroscopic object like a moving ball, the momentum is so large that the de Broglie wavelength is unimaginably small (far smaller than an atomic nucleus), which is why we never observe wave behaviour in everyday objects. For an electron moving at 10⁶ m/s, the momentum is about 9.1×10⁻²⁵ kg·m/s, giving a de Broglie wavelength of about 0.73 nm — comparable to atomic spacings, which is why electron diffraction is used to study crystal structures.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| p | Linear momentum | kg.m/s |
| m | Mass of the object | kg |
| v | Velocity of the object | m/s |
What it means
Linear momentum (p) is a measure of the motion of an object, defined as the product of its mass and velocity. It is a vector quantity, so direction matters. The total momentum of a system is conserved when no external net force acts (conservation of momentum), which is a fundamental principle in physics, equivalent to Newton’s third law. Momentum is used to analyse collisions, explosions, and rocket propulsion. The impulse‑momentum theorem (J = Δp) connects force and time to change in momentum. In everyday life, momentum explains why it is harder to stop a heavy truck than a bicycle. In engineering, momentum is used in fluid mechanics (momentum flux) and in vehicle safety design. Understanding momentum is essential for predicting outcomes of interactions between objects.
Worked example
Linear Momentum – Two Examples
Real‑World| Parameter | Value |
|---|---|
| m | 10 kg |
| v | 5 m/s |
| Parameter | Value |
|---|---|
| m | 1000 kg |
| v | 20 m/s |
Common mistakes
- Mass m: In kg – not weight.
- Velocity v: A vector quantity – momentum is a vector, so direction matters.
- Units: p in kg·m/s.
- Momentum vs. kinetic energy: Momentum is linear (p = mv), energy is scalar (KE = ½mv²) – do not confuse.
- Conservation: In a closed system, total momentum is conserved – but this equation defines momentum of a single object.
Applications
Linear momentum, p = m·v, is a fundamental quantity in mechanics, representing the product of mass and velocity. It is conserved in isolated systems, making it a powerful tool for analysing collisions and explosions. Engineers apply momentum conservation to design vehicle crash safety systems, to analyse rocket propulsion, and to understand fluid impacts. In sports science, momentum explains how mass and velocity affect performance. In astrophysics, it is used to study the motion of celestial bodies. The concept also underpins the design of hydraulic rams, water hammers, and pulse jets. By understanding linear momentum, professionals can predict the outcome of interactions and design systems that harness or mitigate momentum transfer for safety and efficiency.
- Automotive crashworthiness and airbag design
- Rocket thrust and propulsion analysis
- Sports biomechanics and performance optimisation
- Impact and blast load analysis in structures
- Design of momentum‑exchange devices (flywheels, governors)
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