Momentum Calculator

Solve p = mv in any direction, work out the force of an impact from how long it lasted, or run a collision between two objects and see both the sticky and the bouncy outcome. The impact mode is the useful one: the change in momentum is fixed by the crash, the duration is the only thing anyone can engineer, and force falls in exact proportion to it.

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How this momentum calculator works

The first mode is p = mv, solved in any direction. The second is the one that earns the page its keep: given a collision, it works out the force from how long the impact lasted, and shows what the same collision would have done over other durations. The third handles two objects hitting each other, reporting both the perfectly sticky and the perfectly bouncy outcomes and how much energy vanishes in between.

The formula

p = mv
J = FΔt = Δp
sticking together: v = (m1u1 + m2u2) ÷ (m1 + m2)
elastic: v1 = ((m1 − m2)u1 + 2m2u2) ÷ (m1 + m2)

p is momentum in kilogram metres per second, a unit with no special name. J is impulse, which is both the force multiplied by the time it acts and the change in momentum it produces; those are the same thing viewed from two ends. Velocities carry a sign throughout, which is what lets the arithmetic tell a head-on collision from a rear-end one.

Worked example

A 70 kg person in a car at 30 mph. That is 13.41 m/s, so they carry 938.8 kg m/s of momentum. In a crash, all of it has to go somewhere, and the only question is how long it takes.

Stop them in 100 milliseconds, which is roughly what a modern car with a crumple zone and an airbag achieves, and the force is 9,388 N, about 13.7 g. Stop them in 20 milliseconds, which is closer to hitting a rigid barrier in an old car with no crumple zone, and the same person, the same speed and the same momentum produce 46,940 N, about 68 g. Survivable becomes not survivable, and nothing changed except the duration.

That is the entire principle behind every safety device ever built. The momentum change is fixed by the crash. The time is the only variable anyone can engineer, and force falls in exact proportion to it. Crumple zones, airbags, helmet liners, climbing rope stretch, gym mats, bending your knees when you land: every one of them is buying milliseconds.

Why momentum and energy are both needed

They look like they measure the same thing and they do not, and the difference decides real outcomes. Momentum is a vector, so it carries direction and can cancel. Kinetic energy is a scalar that depends on the square of speed, so it is never negative and never cancels.

Put two identical cars head on at the same speed and the total momentum is exactly zero. They stop dead. And yet all of their kinetic energy, which for two 1,500 kg cars at 20 m/s is 600 kJ, still has to go somewhere, and it goes into folding metal. The quantity that cancelled is not the quantity that hurt anyone. Run that case in the collision mode above and both numbers appear side by side.

The division of labour is worth remembering: momentum is always conserved in a collision, whatever happens, which makes it the reliable bookkeeping for working out where things end up. Kinetic energy is only conserved in the idealised elastic case, and the amount that goes missing is the amount that did damage. If you want to know where the pieces go, use momentum. If you want to know how bad it was, use energy.

The one number engineers can actually change

Look again at what is fixed and what is not in a crash. The mass is fixed. The speed is fixed by the moment of impact. So the change in momentum is fixed before anyone gets to design anything. The only free variable in J = FΔt is the duration, and the force follows from it by simple division.

This is why crash safety is a story about distance and time rather than about strength. A stiffer car is not a safer car; a stiffer car stops faster and hits its occupants harder. What a crumple zone does is deliberately fail, over as many centimetres and milliseconds as the packaging allows, so the deceleration is spread instead of concentrated. The same logic explains why a boxer rolls with a punch, why a fielder pulls their hands back as they catch, and why falling onto a mattress and onto concrete deliver identical momentum changes with entirely different consequences.

There is a limit to how far the trick goes, and it is geometric rather than clever. Stretching the stop needs room to stretch it in, and a car only has so much length in front of the passenger compartment. Beyond that, the remaining options are reducing the speed, which is our kinetic energy calculator's territory, and spreading the same force over more of the body, which is what a seatbelt does that a steering wheel does not.

Frequently asked questions

How do I calculate momentum?

Multiply mass by velocity. A 70 kg person travelling at 13.41 m/s (30 mph) carries 938.8 kg m/s. The unit has no special name. Velocity carries a sign, so something moving the other way has negative momentum, and that sign is what lets two momenta cancel.

What is impulse?

The change in momentum, which is equal to force multiplied by the time the force acts. Those two descriptions are the same quantity seen from opposite ends: J equals F times delta t equals delta p. It is the most useful equation in crash safety, because the change in momentum is fixed by the collision while the duration is not.

How do I calculate impact force?

Divide the change in momentum by how long the impact lasted. A 70 kg person going from 13.41 m/s to rest in 100 milliseconds needs 9,388 N, about 13.7 g. Stop the same person in 20 milliseconds and it becomes 46,940 N, about 68 g. The hard part is never the arithmetic, it is knowing the duration, which is why crash engineers measure it rather than estimating it.

Why do crumple zones make cars safer?

Because they stretch out the time. The change in momentum in a crash is fixed by the mass and the speed, so the only variable left is how long the stop takes, and force is momentum change divided by that time. A crumple zone deliberately fails over as many centimetres and milliseconds as the packaging allows, spreading the deceleration instead of concentrating it. A stiffer car is not a safer car: it stops faster and hits its occupants harder.

What is the difference between momentum and kinetic energy?

Momentum is mass times velocity and is a vector, so it has direction and can cancel. Kinetic energy is half mass times speed squared and is a scalar, so it is never negative and never cancels. Two identical cars head on at the same speed have exactly zero total momentum and a great deal of total energy, and it is the energy that folds the metal. Momentum is always conserved, which makes it good bookkeeping; energy conservation fails in real collisions, and the amount that goes missing measures the damage.

Is momentum conserved in every collision?

Yes, as long as no outside force acts during it. Whether the objects bounce, stick together, shatter or explode, the total momentum after equals the total before. That is what makes it useful: you can work out where things end up without knowing anything about what happened during the impact. Kinetic energy has no such guarantee and is only conserved in the idealised perfectly elastic case.

What happens when two objects stick together?

They share the total momentum, so the combined velocity is the sum of the two momenta divided by the sum of the masses. This is a perfectly inelastic collision, and it loses the most kinetic energy of any outcome. Cars are deliberately designed toward this end rather than the bouncy end, because bouncing means a larger velocity change for the occupants and therefore a larger force on them.

Why does landing on a mat hurt less than landing on concrete?

Not because your momentum is any different, because it is identical. The mat compresses, which stretches the stop from a few milliseconds to a few tenths of a second, and the force falls by the same factor. Injury is caused by force, and force is momentum change divided by duration, so the surface changes the only variable in that equation. The same reasoning explains rolling with a punch and bending your knees on landing.

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