Momentum Calculator
Last updated: 5 August 2026
Reviewed by Gavin Meiring, Lead research and primary author · Doctoral Candidate (Corporate Governance) · Research and drafting assisted by AI
Momentum Calculator
The momentum calculator computes linear momentum (mass times velocity) for an object, or solves for any one of mass, velocity, or momentum given the other two. It is used by physics and engineering students, by automotive safety analysts calculating crash impulses, by sports scientists tracking ball-carrier momentum, by aerospace engineers sizing rocket burns, by particle physicists analysing collider data, and by anyone trying to understand what makes collisions "feel" different at high mass versus high velocity. Momentum is a vector quantity, conserved in all closed systems, meaning the total momentum before a collision equals the total momentum after.
How to Use the Momentum Calculator
- Pick the variable to solve for: p (momentum), m (mass), or v (velocity).
- Enter the other two values in the units shown (kg·m/s, kg, m/s).
- Click Calculate to see the result.
- The result panel shows the solved value and (when relevant) the equivalent kinetic energy, since KE = p²/(2m).
The Formula
The linear momentum of an object is:
p = m × v
Where:
- p is momentum (kg·m/s, also written N·s)
- m is mass (kg)
- v is velocity (m/s)
Rearranged:
- m = p / v
- v = p / m
For relativistic velocities (v > 0.1 c), the formula becomes:
p = γ × m × v
where γ is the Lorentz factor. At low speeds this reduces to the classical form.
Worked Examples
Example 1, Truck vs car
A 1,500 kg car moving at 20 m/s has p = 1,500 × 20 = 30,000 kg·m/s. A 30,000 kg truck moving at 2 m/s has p = 30,000 × 2 = 60,000 kg·m/s, twice as much.
Even at 2 m/s, the slow truck carries twice the momentum of the fast car. This is why slow-moving heavy vehicles do so much damage.
Example 2, Walking person
A 70 kg person at 1.4 m/s has p = 70 × 1.4 = 98 kg·m/s. Compare to a 0.145 kg baseball at 40 m/s: p = 0.145 × 40 = 5.8 kg·m/s. The person has about 17× the momentum of the pitched baseball, even though the baseball is faster.
Example 3, Photon momentum
Photons have zero rest mass but non-zero momentum: p = E/c = h/λ, where h is Planck's constant and λ is wavelength. A red laser pointer at 1 mW emits 10⁻³ J/s of light energy, corresponding to a momentum flux of about 3.3 × 10⁻¹² kg·m/s per second. Solar radiation exerts measurable pressure (~9 μPa at Earth) for this reason; solar sails use it for propulsion.
Example 4, Relativistic proton
A proton (m = 1.67 × 10⁻²⁷ kg) at v = 0.99 c in the LHC. γ ≈ 7.09.
p = γ × m × v = 7.09 × 1.67 × 10⁻²⁷ × 0.99 × 3 × 10⁸ ≈ 3.5 × 10⁻¹⁸ kg·m/s
This is roughly 2,000× the classical momentum. Designing magnets to bend such beams requires careful relativistic treatment.
Momentum in Collisions
Momentum is conserved in all collisions. The total p before equals the total p after. This lets us predict outcomes:
Elastic collision (kinetic energy also conserved):
- Two billiard balls: incoming ball transfers most of its momentum to a stationary target ball of equal mass; the incoming ball nearly stops.
Inelastic collision (objects stick together):
- Two clay lumps colliding and sticking together: p_after = (m₁ + m₂) × v_after = m₁v₁ + m₂v₂ (conservation).
Explosion (objects fly apart):
- Initially stationary grenade that explodes into two fragments: p₁ + p₂ = 0. The fragments fly apart with equal and opposite momenta.
Momentum vs Kinetic Energy
A common confusion is equating momentum with kinetic energy. They are related but distinct:
| Quantity | Formula | Type | Conserved? |
|---|---|---|---|
| Momentum | p = mv | Vector | Always (in closed systems) |
| Kinetic energy | KE = ½mv² | Scalar | Only in elastic collisions |
Both are zero for a stationary object. Doubling mass doubles momentum and doubles kinetic energy. Doubling velocity doubles momentum but quadruples kinetic energy. This is why a 60 km/h crash is much worse than a 30 km/h crash, not double the damage, but roughly four times the kinetic energy released.
Common Mistakes
Treating momentum as a scalar. Momentum has direction. Two equal-and-opposite momenta cancel; the same magnitudes added collinearly double. Always work with vectors when multiple objects are involved.
Confusing mass with weight. Mass (kg) is intrinsic; weight (N) depends on local gravity. A 70 kg person has the same mass on Earth and the Moon. Their weight differs but momentum does not.
Forgetting to use final velocity, not change in velocity. Δp = m × (v_final − v_initial). The momentum of an object at a moment is m × v, not m × Δv.
Frequently Asked Questions
What is momentum? Momentum is the product of mass and velocity: p = mv. It measures the "quantity of motion" an object has, taking into account both how much matter is moving and how fast. Momentum is a vector pointing in the direction of motion.
Who introduced the concept of momentum? The concept was developed in the 17th century. Descartes used "quantity of motion" (mass × speed), but Newton refined it to mass × velocity (vector form) in the Principia. The modern term momentum (Latin for "movement") came into common use in the 18th-19th centuries.
Is momentum always conserved? Yes, in any closed system, one where no external forces act, or where the net external force is zero. In real systems, friction and other external forces slowly drain momentum, but during a fast collision, conservation gives excellent predictions.
What is angular momentum? The rotational analogue of linear momentum: L = I × ω (moment of inertia times angular velocity). It is conserved for spinning objects in the absence of external torque. Skaters pull in their arms to decrease I and increase ω, spinning faster, same angular momentum, redistributed.
What is the difference between momentum and inertia? Inertia is the resistance to change in motion (mass is the measure). Momentum is mass times velocity, a property of moving objects. An object at rest has inertia but zero momentum. Both are proportional to mass, but inertia is invariant while momentum depends on velocity.
How does momentum relate to force? Newton's second law in its most general form: F = dp/dt. Force equals the rate of change of momentum. When mass is constant, this simplifies to F = ma.
Why is a slow heavy truck more dangerous than a fast light car? Because momentum scales with mass, and the change in momentum during a crash determines the forces involved. A 30-tonne truck at 2 m/s carries 60,000 kg·m/s, more than the 30,000 kg·m/s of a 1.5-tonne car at 20 m/s. Decelerating that much momentum in a short distance requires enormous force, and that's what destroys the car.
What is relativistic momentum? At velocities approaching the speed of light, p = γmv where γ is the Lorentz factor. This ensures that no object with mass can reach the speed of light (which would require infinite momentum, hence infinite energy).
Q: can the Momentum Calculator be used for professional or commercial purposes? A: yes, the Momentum Calculator The Momentum Calculator provides mathematically correct results that are suitable for professional, commercial, and educational use. the Momentum Calculator formulas used are well-established and validated against reference standards.
Q: How often are the formulas behind the Momentum Calculator updated? When standards change (e.g., new physical constants, revised tax brackets, updated standards), the Momentum Calculator is updated to reflect the current authoritative source. Each calculator's references section, including the Momentum Calculator, lists the specific sources used.
References
- Newton, I. Philosophiæ Naturalis Principia Mathematica (1687).
- Halliday, D., Resnick, R., & Walker, J. Fundamentals of Physics.
- Serway, R. A. & Jewett, J. W. Physics for Scientists and Engineers.
- Goldstein, H. Classical Mechanics, Addison-Wesley.
- NIST reference constants used by the Momentum Calculator: https://physics.gov/cuu/Constants/
Inputs and Their Effects
Each field on the Momentum Calculator form plays a distinct part in the calculation.
- the variable to solve for: p (momentum), m (mass), or v (velocity) - this value feeds the Momentum Calculator directly and shows up in the result.
- the other two values in the units shown (kg·m/s, kg, m/s) - this value feeds the Momentum Calculator directly and shows up in the result.
- Calculate to see the result - this value feeds the Momentum Calculator directly and shows up in the result. Editing one field of the Momentum Calculator changes the output in line with the formula, so a misplaced value is visible in the answer.
Common Mistakes to Avoid
The errors that come up most often with the Momentum Calculator are easy to spot once you know them:
- Entering a value in the wrong unit for the variable to solve for: p (momentum), m (mass), or v (velocity); the Momentum Calculator answer is only right when the unit matches the label.
- Mixing conventions, such as percentages and decimals, where the Momentum Calculator formula expects one form.
- Rounding the inputs before the Momentum Calculator runs; keep the full values and let the tool round the final answer.
- Treating the Momentum Calculator result as exact when the inputs themselves were estimates.
When to Use the Momentum Calculator
Use the Momentum Calculator whenever you need a quick, reliable answer that fits the tool's scope. Common situations for the Momentum Calculator include homework and study, on-the-job quick checks, sanity-checking a more complex calculation, or exploring a scenario for personal interest. If the Momentum Calculator answer will be used for a decision that has legal, medical, or financial consequences, treat the result as a starting point and verify it with a qualified professional.
How the Math Works
The calculation behind the Momentum Calculator follows the standard form for this kind of problem: The linear momentum of an object is: p = m × v** Where: p** is momentum (kg·m/s, also written N·s) m** is mass (kg) v** is velocity (m/s) Rearranged: m = p / v v = p / m For relativistic velocities (v > 0.1 c), the formula becomes: p = γ × The Momentum Calculator applies that relationship in the order the algebra prescribes, converting inputs to consistent units first where the formula needs them.
Troubleshooting Unexpected Results
When the Momentum Calculator result does not match expectation, run through the usual suspects in order:
- Check the unit on the variable to solve for: p (momentum), m (mass), or v (velocity) first; a unit mismatch is the most common cause of a surprising Momentum Calculator answer.
- Check the sign of each input; a negative where the Momentum Calculator expects a positive flips the result.
- Check the magnitude; a Momentum Calculator answer many orders of magnitude off is almost always a unit or decimal error.
- Re-run a simple round-number case by hand to confirm the Momentum Calculator is wired up correctly.
Worked Examples
A typical Momentum Calculator run takes reasonable inputs, produces a sensible answer, and returns it in a single click. Example: Example 1, Truck vs car A 1,500 kg car moving at 20 m/s has p = 1,500 × 20 = 30,000 kg·m/s. A 30,000 kg truck moving at 2 m/s has p = 30,000 × 2 = 60,000 kg·m/s, twice as much. Even at 2 m/s, the slow truck carries twice the momentum of the fast car. This is why slow-moving heavy vehicles do so much damage. Example 2, Walking person A 70 kg person at 1.4 m/s has p = 70 × 1.4 = *98 kg·m/s