The momentum calculator above solves p = mv in whichever direction you need it. Give it a mass and a velocity for the momentum, a momentum and a velocity for the mass, or a momentum and a mass for the velocity. It also reports the kinetic energy of the same object, the impulse needed to stop it, and the average force that impulse implies over a time interval you choose.
Arb Digital builds free tools that pick one job and finish it properly. This page handles a single moving body. The boundary against the site's existing collision momentum calculator is worth stating plainly, because the two sound similar and are not: that page takes two bodies and applies conservation of momentum through an impact to find what happens afterwards. This page never involves a second body. It is the definition of momentum for one object, rearranged.
What This Momentum Calculator Does
Its core job is the product of mass and velocity, but a bare product would be a thin tool. What makes momentum useful is the comparison with the quantities that sit next to it, so the result panel puts four of them in view at once.
Kinetic energy is the first. It is half the mass times the velocity squared, and because of that square it behaves quite differently. Momentum is a vector and is conserved in every collision. Kinetic energy is a scalar and is conserved only in perfectly elastic ones. When a car crumples, momentum is conserved exactly and energy is not, and that difference is the whole reason crumple zones work.
The other three figures answer the practical question that usually follows. Impulse is the change in momentum, so bringing an object to rest requires an impulse equal to its momentum. Divide that by a stopping time and you have the average force involved. Multiply the average velocity by the same time and you have the distance covered while stopping, which is the deformation the structure has to provide.
How to Use It
- Pick the quantity you want. Momentum is the usual case. Solving for mass suits problems where a momentum is measured and the object is unknown, and solving for velocity suits recoil and ejection problems.
- Enter velocity in whatever unit you have. The selector converts kilometres per hour, miles per hour and feet per second into metres per second before anything is calculated, because the SI unit is the only one the formulas accept.
- Use magnitudes, not signed values. Momentum is a vector and direction matters in a real problem, but this page computes magnitude. For opposing directions in a collision, use the collision tool instead.
- Set a realistic stopping time. This is the input people guess worst. It is the duration of the deceleration, not the reaction time before it and not the time to come to a halt under braking.
- Compare the force figure against the energy figure. They answer different questions, and a change that halves one does not halve the other.
The Formula: How Momentum Is Calculated
Linear momentum is p = mv, the product of mass and velocity. OpenStax University Physics Volume 1, section 9.1 on linear momentum, states that the momentum of an object is the product of its mass and its velocity, and treats it as a vector quantity pointing along the velocity. The coherent SI unit is the kilogram metre per second, identical in dimensions to the newton second.
Rearranging gives the other two forms: m = p ÷ v and v = p ÷ m. Both are guarded here against a zero denominator, because a stationary object has zero momentum for any mass and the division tells you nothing.
Work the default values. A 1,500 kilogram car at 20 metres per second has a momentum of 1,500 × 20 = 30,000 kilogram metres per second. Its kinetic energy is 0.5 × 1,500 × 202 = 300,000 joules, or 300 kilojoules. Stopping it requires an impulse of 30,000 newton seconds, so over 0.15 seconds the average force is 30,000 ÷ 0.15 = 200,000 newtons, or 200 kilonewtons. The distance covered while decelerating uniformly is the average velocity times the time, 10 × 0.15 = 1.5 metres.
Momentum Against Kinetic Energy
These two quantities are so often confused that it is worth being explicit about where they differ. Both are built from mass and velocity. Momentum uses velocity to the first power and is a vector; kinetic energy uses velocity squared and is a scalar. That single difference produces almost every counter-intuitive result in mechanics.
Consider a 10 gram bullet at 400 metres per second and a 4 kilogram bowling ball at 1 metre per second. Both have a momentum of 4 kilogram metres per second, so both would deliver the same impulse and, given the same stopping time, the same average force. Their kinetic energies are 800 joules and 2 joules respectively, a ratio of four hundred to one. Momentum says they are equivalent; energy says they are not remotely equivalent. The energy figure is the one that predicts damage, because damage is work done.
The same logic explains recoil. A rifle and its bullet leave with equal and opposite momentum, which is why recoil momentum equals bullet momentum exactly. But the rifle is hundreds of times heavier, so its velocity is hundreds of times smaller, and its kinetic energy — scaling with the square of that velocity — is hundreds of times smaller again. Equal momentum, wildly unequal energy. The kinetic energy calculator handles the energy side in more depth.
Impulse and Why Stopping Time Matters So Much
The impulse-momentum theorem says that force multiplied by the time it acts equals the change in momentum. Turn that around and it becomes the most practically important statement in vehicle safety: for a given change in momentum, the longer the deceleration takes, the smaller the force.
The momentum change is fixed by the mass and the speed. Nothing about the vehicle design can alter it. What design can alter is the time over which it happens. A rigid structure hitting a barrier stops in perhaps 0.05 seconds; a crumple zone can stretch that to 0.15 or more. Tripling the time divides the average force by three, and it is the force on the occupants that injures them.
Try it in the tool. Leave the car at 20 metres per second and change the stopping time from 0.05 to 0.15 seconds: the momentum and kinetic energy do not move at all, while the average force falls from 600 kilonewtons to 200. The stopping distance rises from 0.5 metres to 1.5, which is precisely the crush depth the structure must supply. This is also the reason airbags, helmet liners and landing mats work — each one buys time. The stopping distance calculator covers the braking case, where the deceleration is far gentler and the distances far longer.
Why Momentum Is Conserved and Energy Is Not
Conservation of momentum follows directly from Newton's third law. OpenStax University Physics Volume 1, section 5.5 on Newton's third law, states that whenever one body exerts a force on a second, the first experiences a force equal in magnitude and opposite in direction. During a collision those two forces act for exactly the same duration, so the two impulses are equal and opposite, so the two momentum changes cancel and the total is unchanged.
Kinetic energy has no such guarantee. Energy is conserved overall, but it can leave the mechanical account entirely — into permanent deformation of metal, into heat, into sound. A collision where kinetic energy is preserved is called elastic and is rare outside billiard balls and gas molecules. One where the objects move off together is perfectly inelastic and loses the most.
This asymmetry is why momentum is the right tool for predicting post-collision velocities and energy is the right tool for predicting damage. Using energy conservation to find velocities after a real crash gives an answer that is simply wrong, because the energy went somewhere the equation does not track.
Momentum at Speeds Where Relativity Bites
The formula p = mv is a low-speed approximation. The exact expression includes the Lorentz factor, so momentum grows without limit as velocity approaches the speed of light while velocity itself never reaches it. At everyday speeds the correction is invisible — at 300 metres per second, roughly Mach 1, it is about one part in two hundred billion.
The correction becomes noticeable above roughly ten per cent of light speed, where it exceeds half a per cent, and dominates above eighty per cent. Nothing this calculator is likely to be used for goes anywhere near that, but it is worth knowing the classical formula has a domain. The time dilation calculator and the mass energy equivalence calculator cover the relativistic regime.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Mixing velocity units — the formula needs metres per second. Entering kilometres per hour without selecting that unit overstates momentum by a factor of 3.6.
- Using weight where mass belongs — a figure in newtons or pounds-force is a weight and must be divided by gravitational acceleration before it can be used as a mass.
- Treating momentum as a measure of damage — damage tracks energy, which scales with the square of velocity. Equal momentum does not mean equal destructive effect.
- Guessing the stopping time from the braking time — impulse figures assume the deceleration you specify, and a crash lasting a tenth of a second is a completely different problem from a stop lasting three seconds.
- Ignoring direction in a real collision — momentum is a vector, and two objects approaching head-on have momenta that partly cancel. This page gives magnitudes only.
Related Free Tools From Arb Digital
For two-body impacts use the collision momentum calculator, which applies conservation through the impact rather than defining momentum for one object. The kinetic energy calculator and the work calculator cover the energy side, the force calculator and the acceleration calculator cover Newton's second law, and the velocity calculator and stopping distance calculator cover the motion itself. Rescale inputs with the speed converter. The full free online tools hub lists everything Arb Digital publishes.
Frequently Asked Questions
Momentum equals mass multiplied by velocity, written p = mv. Mass is in kilograms and velocity in metres per second, giving momentum in kilogram metres per second. Rearranging gives m = p divided by v, and v = p divided by m.
No. Momentum uses velocity to the first power and is a vector; kinetic energy uses velocity squared and is a scalar. A bullet and a bowling ball can have identical momentum while their kinetic energies differ by a factor of several hundred.
The kilogram metre per second, which is the coherent SI unit. It has the same dimensions as the newton second used for impulse, which is why an impulse in newton seconds can be added directly to a momentum in kilogram metres per second.
That tool takes two bodies and applies conservation of momentum through an impact to find the velocities afterwards. This page involves a single body and simply rearranges the definition p = mv, adding the energy and impulse figures that go with it.
Because impulse equals force multiplied by time, and the impulse required is fixed by the momentum change. Spreading the same impulse over three times the duration divides the average force by three, which is exactly what a crumple zone or an airbag is designed to do.
Yes, in a real problem, because it is a vector and the sign records direction along a chosen axis. This page reports magnitudes, so for objects moving in opposite directions you need a tool that tracks sign, such as the collision momentum calculator.
Only as an approximation. The exact relativistic expression multiplies by the Lorentz factor, which is negligible below about a tenth of the speed of light and dominant above eighty per cent of it. Everyday problems are comfortably inside the classical range.
This tool is provided for educational and estimating use. The force and stopping figures assume uniform deceleration over the time you enter, and nothing on this page is engineering, safety or vehicle design guidance.
