Momentum is the “quantity of motion” an object carries: its mass multiplied by its velocity. It explains why a 20-tonne truck creeping at 5 km/h (about 27,800 kg·m/s) is nearly as hard to stop as a 1,500 kg car at 72 km/h (30,000 kg·m/s). This calculator solves p = mv for whichever of the three values you are missing, keeps track of direction with signs, and also reports the object’s kinetic energy.
How to use the momentum calculator
- Under Solve for, pick Momentum, Mass or Velocity.
- Enter the two known values and choose their units. Mass accepts kg, g, mg, metric tonnes (t), lb, oz, US short tons and stone; velocity accepts m/s, km/h, mph, ft/s and knots; momentum accepts kg·m/s, N·s, g·cm/s and lb·ft/s.
- Give velocity and momentum a sign: positive for the direction you chose as forward, negative for the opposite one.
- Use Show the result in to pick the output unit. The result also lists other units, the kinetic energy (p² ÷ 2m) and the average force that would stop the object in one second, followed by the worked steps.
Momentum formula
Rearranged for the other two unknowns:
Every input is converted to SI first (kilograms and meters per second), so the product comes out in kg·m/s. Mass must be greater than zero. When solving for mass, neither the momentum nor the velocity can be zero (a resting object has no momentum to divide), and momentum and velocity must share a sign, otherwise the answer would be a negative mass and the calculator rejects it.
Worked example: a car at 72 km/h
A 1,500 kg car travels at 72 km/h.
Convert the speed: 72 ÷ 3.6 = 20 m/s.
Momentum: p = 1,500 kg × 20 m/s = 30,000 kg·m/s, about 216,990 lb·ft/s.
Kinetic energy: 30,000² ÷ (2 × 1,500) = 300,000 J.
Stopping it in one second takes an average force of 30,000 N.
In US units the arithmetic is just as direct: a 3,300 lb car at 45 mph (exactly 66 ft/s) carries 3,300 × 66 = 217,800 lb·ft/s, or about 30,112 kg·m/s.
Momentum is a vector
Speed has no direction, but velocity and momentum do. On a straight road, choose one direction as positive and stick with it; anything moving the other way gets a negative velocity, and the calculator returns a negative momentum to match. In two or three dimensions, split the motion into components and handle each axis separately, since momentum is conserved along every axis independently.
Conservation of momentum: a head-on collision
When no outside force acts during an impact, the total momentum of the objects before the collision equals the total afterward. Take car A from the example above (1,500 kg, +72 km/h) and car B, a 1,000 kg hatchback coming the other way at 54 km/h:
- Car A: +30,000 kg·m/s
- Car B: 1,000 kg × (−15 m/s) = −15,000 kg·m/s
- Total before impact: +15,000 kg·m/s
If the cars lock together, the wreck has a mass of 2,500 kg and still carries +15,000 kg·m/s. Solving for velocity with those numbers gives +6 m/s (21.6 km/h), moving in car A’s original direction. Kinetic energy tells a different story: 300,000 J + 112,500 J = 412,500 J before, only 45,000 J after. About 89% of the energy went into crushing metal, heat and noise. Momentum survives a crash intact; kinetic energy does not. Only in elastic collisions, such as near-perfect bounces between billiard balls, are both conserved.
Momentum units
| Unit | Value in kg·m/s | Typical use |
|---|---|---|
| kg·m/s | 1 | SI unit |
| N·s | 1 (identical) | Impulse, rocket thrust |
| g·cm/s | 0.00001 | CGS system, older texts |
| lb·ft/s | 0.138254954376 | US engineering (pound-mass) |
Watch the pound: lb·ft/s uses the pound as a mass, while the lbf·s found in impulse work uses the pound-force. One lbf·s equals about 32.174 lb·ft/s, the same factor as standard gravity in ft/s².
Related calculators
To find how much push changes an object’s momentum, use the impulse calculator or solve the full theorem with the impulse-momentum calculator. For energy at a given speed, see the kinetic energy calculator, and to switch between momentum units, the momentum converter.
Frequently asked questions
What are the units of momentum?
The SI unit is the kilogram meter per second (kg·m/s), which is the same thing as a newton second (N·s). In US engineering work you will also meet the pound foot per second (lb·ft/s), equal to exactly 0.138254954376 kg·m/s. The calculator reads and reports kg·m/s, N·s, g·cm/s and lb·ft/s.
Can momentum be negative?
Yes. Momentum is a vector, so along a straight line its sign shows direction. If you call eastward positive, a car heading west has negative momentum. Mass is never negative, so momentum always has the same sign as the velocity.
Is momentum conserved when two cars crumple in a crash?
Yes, provided outside forces such as tire grip are small compared with the impact forces during the brief collision. Kinetic energy is not conserved in that case, because much of it goes into bending metal, heat and sound, but the total momentum just after the impact matches the total just before.
How are momentum and kinetic energy related?
For one object, KE = p² ÷ 2m. Equal momentum does not mean equal energy: a 10 g bullet at 400 m/s and a 4 kg object at 1 m/s both carry 4 kg·m/s, yet the bullet has 800 J of kinetic energy and the heavy object only 2 J.
What does the 'force to stop it in 1 s' line mean?
It is the average force that would bring the object to rest in exactly one second, which is numerically equal to the size of its momentum. Halving the stopping time doubles the force. For other stopping times, use the impulse calculator.