Momentum Calculator

Momentum is p = m × v: mass multiplied by velocity. Choose which quantity to solve for above and the calculator rearranges the equation and converts between kilogram metres per second and other units.

What momentum measures

Momentum captures how hard something is to stop. A quantity that combines mass and velocity is more useful than either alone, because a heavy object moving slowly and a light one moving quickly can be equally difficult to arrest.

A 1500 kg car at 20 m/s carries 1500 × 20 = 30,000 kg·m/s. An 8 gram bullet at 900 m/s carries 0.008 × 900 = 7.2 kg·m/s. The car has more than four thousand times the momentum of the bullet despite moving at a small fraction of its speed, which is why a car is far harder to stop than a rifle round.

Momentum is a vector, so direction is part of it. Two objects of equal mass moving toward each other at the same speed have equal and opposite momentum, and their total is zero even though neither is at rest.

A worked example

Take a mass of 1500 kg moving at 20 m/s.

• Multiply mass by velocity: 1500 × 20 = 30,000 kg·m/s

Double the velocity to 40 m/s and the momentum doubles to 60,000. Momentum is linear in speed, unlike kinetic energy which is quadratic. That difference matters, and it is covered below.

Conservation, and why it is so useful

In any collision, the total momentum before equals the total momentum after, provided no outside force acts. This holds whether the objects bounce apart, stick together or shatter, which makes it the single most reliable tool for analysing impacts.

Take a 1500 kg car at 20 m/s striking a stationary 1000 kg car and locking together. Total momentum before is 30,000 kg·m/s. After the collision the combined 2500 kg mass carries the same 30,000, so its velocity is 30,000 / 2500 = 12 m/s.

Notice that kinetic energy is not conserved here. Before the crash the moving car carried 300 kJ; afterwards the pair carry ½ × 2500 × 144 = 180 kJ. The missing 120 kJ went into deforming metal, heat and noise. Momentum survives the collision; energy does not.

Impulse: changing momentum takes time

The change in momentum equals force multiplied by the time it acts, a quantity called impulse. Written out, FΔt = Δp.

That equation explains a large amount of safety design. To bring a given momentum to zero you need a fixed impulse, but you can choose how to supply it. A large force over a short time or a small force over a long one both work, and the second is survivable.

A 70 kg occupant at 15 m/s carries 1050 kg·m/s. Stopping that in 0.3 seconds needs 3500 N. Stopping it in 0.03 seconds needs 35,000 N. The momentum to be removed is identical; only the time differs.

The same reasoning is why a cricketer draws their hands back while catching, why a boxer rolls with a punch, and why crash barriers are designed to deform rather than resist.

Elastic and inelastic collisions

Type	Momentum	Kinetic energy	Example
Elastic	conserved	conserved	billiard balls, gas molecules
Inelastic	conserved	partly lost	most real collisions
Perfectly inelastic	conserved	maximum loss	objects that stick together

Perfectly elastic collisions are rare above the molecular scale. Even a good snooker shot loses a few percent of its energy to sound and friction, and the ideal case exists mainly as a limiting assumption in problems.

Recoil

Fire a bullet and the total momentum must stay at zero, since it was zero before the trigger was pulled. If an 8 g bullet leaves at 900 m/s with 7.2 kg·m/s, a 4 kg rifle must recoil at 7.2 / 4 = 1.8 m/s in the opposite direction.

The rifle is 500 times the mass, so it moves 500 times slower, which is what makes the recoil bearable. The same arithmetic governs rocket propulsion, where the expelled gas carries momentum one way and the vehicle carries it the other.

Where it gets used

Crash investigation. Working backwards from post-impact positions and masses reconstructs pre-impact speeds, because momentum is conserved even when energy is not.

Sport. Follow-through extends the time a force acts, which increases the impulse and so the momentum given to the ball.

Particle physics. Conservation of momentum is how unseen particles are inferred: if the momentum does not balance, something left the detector unrecorded.

Spacecraft. With nothing to push against, changing velocity means throwing mass overboard and letting conservation do the rest.

Why heavy vehicles need so much more room

Momentum is the reason a lorry cannot stop like a car. A 40 tonne articulated vehicle at 25 m/s carries 40,000 × 25 = 1,000,000 kg·m/s, against 37,500 for a 1500 kg car at the same speed. That is nearly 27 times as much to remove.

Braking force is limited by tyre grip, which rises with weight but not by enough to compensate, and heavy vehicles also take longer to build up brake pressure. The result is a stopping distance roughly double that of a car from the same speed, which is why pulling in front of a lorry and braking is so dangerous.

Angular momentum

Spinning objects have a rotational equivalent, angular momentum, which is conserved in the same way. It equals the moment of inertia multiplied by the rotation rate.

A skater pulling their arms in reduces their moment of inertia, so the rotation rate must rise to keep the product constant. Nothing pushes them faster; the redistribution of mass does it. The same conservation keeps a bicycle upright, stabilises a spinning bullet and explains why a cat can turn in mid-air without anything to push against.

Units, and what they tell you

Momentum is measured in kilogram metres per second, and unlike most quantities it has no named unit of its own. That is a small clue to its nature: it is a compound of mass and motion rather than something measured directly.

The unit is identical to the newton second, which is the unit of impulse, and that is not a coincidence. Impulse is a change in momentum, so the two must share units. Seeing N·s and kg·m/s used interchangeably in a textbook is correct rather than sloppy.

Impulse: the same change, spread over time

Changing momentum takes a force applied for a time, and the product of the two is the impulse. Since the change is fixed by the starting and finishing speeds, a longer contact time means a smaller force.

A car stopping from 25 m/s must shed 37,500 kg·m/s whatever happens. Hit a wall and that occurs in perhaps 0.1 seconds, giving an average force of 375,000 N. A crumple zone stretching the same change to 0.4 seconds cuts the force to 93,750 N. Airbags, helmet foam and a cricketer's hands moving back with the catch all work this way, and none of them reduces the momentum. They only lengthen the time over which it goes.

Common mistakes

Ignoring direction. Momentum is a vector. Opposing motions subtract rather than add.

Confusing it with kinetic energy. Momentum is mv and energy is ½mv². Doubling speed doubles one and quadruples the other.

Assuming energy is conserved too. In most collisions it is not. Only momentum survives intact.

Mixing units. Kilograms and metres per second, or convert first.