Momentum Calculator - p = mv, Impulse and Collisions

Mass in motion, and the one quantity a crash cannot destroy

A 1,400 kg car at 50 km/h, 31 mph, carries 19,444 kg m/s of momentum. Nothing that happens in the next second can make that number disappear: it can be handed to another body, to the road, to the planet, but it cannot be destroyed. Kinetic energy has no such protection, which is why a bumper crumples. This calculator works in three modes: momentum of a single body from mass and velocity, the impulse that changes it, and a one-dimensional collision between two bodies with the before and after laid out side by side.

Masses go in as kilograms, grams, metric tons or pounds; velocities as meters per second, km/h or mph. Everything is converted to SI before any arithmetic runs, and the conversion is shown so you can see what the calculator actually worked with. A worked case: 0.43 kg soccer ball struck at 30 m/s, 67 mph, gives 12.9 kg m/s of momentum and 193.5 J of kinetic energy.

One calculation from the first field to the answer

What you want to work out - momentum of one body, the impulse acting on it, or a collision between two. The form changes to match; nothing irrelevant to the chosen mode stays on screen.
Mass unit - kilograms for most things, grams for a bullet or a ball, metric tons for a truck or a train, pounds if that is what the specification sheet says. One pound is exactly 0.45359237 kg.
Speed unit - meters per second is the physics unit, km/h and mph are what speedometers show. One mph is exactly 0.44704 m/s; km/h divides by 3.6.
Mass and velocity - in momentum mode, the two numbers the formula needs. A minus sign on the velocity means the body travels the other way along the same line, and the momentum comes out negative with it.
Force and contact time - in impulse mode. The product is the impulse, and the calculator also shows the same impulse spread over ten times longer and ten times shorter, which is the whole argument for crumple zones in one table.
Both masses and both velocities - in collision mode, with a sign on each velocity. Two bodies approaching head on take opposite signs; a rear-end shunt takes the same sign on both.
Kind of collision - perfectly inelastic if the bodies end up moving together, elastic if they separate cleanly. Real collisions sit between the two, and these are the bounds.
Contact time, optional - give it and each body's change in momentum is divided by it to get the average force during contact. Leave it out and the collision is still solved, just without the force.
Read the results - the tiles hold the headline figures, the table holds momentum and energy for each body before and after, and the last line always states the momentum total both sides of the impact so you can see it has not moved.

Elastic against perfectly inelastic, side by side

Both kinds conserve momentum. They differ on energy, and that single difference decides the velocities that come out. The table below compares them on the same pair of bodies: 1,200 kg at 20 m/s, 45 mph, running into 1,500 kg at -15 m/s, 34 mph the other way, head on.

What is compared	Perfectly inelastic	Elastic
What happens to the bodies	They stay together	They separate
Momentum	Conserved	Conserved
Kinetic energy	Partly lost	Conserved
Velocity after, body 1	0.5556 m/s	-18.889 m/s
Velocity after, body 2	0.5556 m/s	16.111 m/s
Energy lost	408,333.3 J, 99.9 percent	0 J
Where you meet it	Cars that lock together, clay, a catch	Pool balls, gas molecules, a hard bounce

Look at the elastic row again. The lighter body arrived at +20 m/s and leaves at -18.889 m/s: it has turned around and kept almost all of its speed. That is not a bug, and it is not energy appearing from nowhere. What an elastic collision reverses is the closing speed, not each body's ground speed. The two were approaching at 20 minus -15, so 35 m/s; afterwards they separate at 16.111 minus -18.889, which is 35 m/s again. That reversal is the whole content of the word elastic.

The three formulas, worked through with numbers

p = m × v
J = F × t = Δp
m1·v1 + m2·v2 = m1·u1 + m2·u2

Example 1, a car in town. Mass 1,400 kg, speed 50 km/h or 31 mph. First the unit: 50 divided by 3.6 is 13.89 m/s. Then p = 1,400 × 13.89.
Result: 19,444.4 kg m/s, and 135,030.9 J of kinetic energy.

Example 2, the same car stopped by brakes. To take that momentum to zero the brakes must supply an impulse of 19,444.4 N s. Over 4 s of braking that is 19,444.4 divided by 4.
Result: an average force of 4,861.1 N, which is about 0.35 g of deceleration.

Example 3, the same car stopped by a wall. Same 19,444.4 N s, but the stop lasts 0.12 s instead of four seconds.
Result: 162,036.7 N, thirty-three times the braking force. The impulse never changed; only the time did.

Example 4, a rear-end shunt that locks. 1,200 kg at 20 m/s into 1,500 kg standing still, bumpers interlocked. Total momentum 24,000 kg m/s, shared over 2,700 kg.
Result: both move off at 8.889 m/s. Of the original 240,000 J, only 106,666.7 J is still motion, so 133,333.3 J went into bent metal.

Example 5, equal masses on a pool table. 0.16 kg cue ball at 5 m/s, 11 mph, into an identical ball at rest, elastic.
Result: the cue ball stops dead and the target leaves at 5 m/s. Equal masses in an elastic head-on collision swap velocities exactly.

Example 6, a ball against a wall. 1 kg at 10 m/s, 22 mph, into 1,000 metric tons at rest, elastic.
Result: the ball comes back at -10.000 m/s and the wall picks up 2 × 10^-5 m/s. The wall really does move; the number is just too small to matter.

What a given momentum actually looks like

A figure in kg m/s means nothing on its own until you have something to hang it on. These are the reference points the calculator uses to place your result, and each one is a mass multiplied by a speed, so the arithmetic can be checked rather than trusted.

Body	Mass	Speed	Momentum
Tennis serve	58 g	60 m/s 134 mph	3.5 kg m/s
Rifle bullet	9.5 g	830 m/s 1,857 mph	7.9 kg m/s
Struck soccer ball	0.43 kg	30 m/s 67 mph	12.9 kg m/s
Sprinter at full speed	80 kg	10 m/s 22 mph	800 kg m/s
Car in town	1,400 kg	13.9 m/s 31 mph	19,460 kg m/s
Loaded truck on the highway	12 t	22.2 m/s 50 mph	266,400 kg m/s
Freight train	2,000 t	30 m/s 67 mph	60,000,000 kg m/s

Two rows in that table are worth sitting with. A rifle bullet and a tennis serve sit within about a factor of two of each other, 7.9 against 3.5 kg m/s, which is close enough to be surprising. What separates them is not momentum but energy: the bullet carries 3,272 J to the serve's 104 J, and it delivers that through a few square millimeters instead of a racket face. And the sprinter beats the bullet by a factor of a hundred, which is the honest answer to why nobody is knocked backwards through a window by being shot.

Where the arithmetic usually goes wrong

Dropping the minus sign on a head-on collision. Two cars approaching each other at 20 and 15 m/s, 45 and 34 mph, do not add up to 35. One velocity is negative, and if both go in positive the calculator will happily solve a rear-end shunt instead. The sign is the direction, and momentum is the quantity where direction is not decoration.

Feeding km/h into a formula that wants m/s. The most common single error in this topic, and it is not a small one: it inflates momentum by a factor of 3.6 and kinetic energy by nearly 13. Set the unit selector and let the conversion happen where it can be seen.

Expecting kinetic energy to be conserved in a crash. It is not, and it must not be. In the head-on example above, 99.9 percent of it disappears into bent steel. Only momentum survives every collision; energy survives only the elastic ones.

Treating a heavier car as automatically safer. Its own occupants do better, because the heavy body changes velocity less. But the total momentum of both cars is unchanged, so whatever the heavy car avoids is handed to the light one. The quantity is conserved, not created, and someone absorbs it.

Reading a large impulse as a large force. They are different quantities separated by time. A parachutist lands with a large impulse and a survivable force; the same impulse in a tenth of the time is a fatal one. If a number in newton seconds is being discussed as though it were newtons, something has gone wrong.

Six questions the formula does not answer on its own

Is momentum the same thing as force?

No, and the units say so. Momentum is kg m/s, a property a moving body has. Force is N, something acting on it. They meet through time: a force applied for a while produces an impulse in N s, and one newton second is one kilogram meter per second. Force is the rate at which momentum changes, not momentum itself.

Why does the calculator give a negative momentum?

Because you entered a negative velocity, and it kept the sign. In one dimension, direction is a plus or a minus, and the sign has to survive the multiplication or a head-on collision cannot be solved at all. Two bodies at +20 and -15 m/s have a total momentum far smaller than either on its own, and that is the physically correct answer.

Which kind of collision should I pick if I do not know?

Run both. Real collisions fall between the two, so perfectly inelastic and elastic give you the bounds rather than a guess. If the bodies visibly separate afterwards, elastic is the closer bound; if they end up tangled or moving as one, inelastic is. Vehicles sit much nearer the inelastic end: the coefficient of restitution quoted in accident reconstruction for car bodywork at impact speeds above walking pace is commonly 0.1 to 0.3, where 0 is perfectly inelastic and 1 is perfectly elastic.

Does this handle collisions at an angle?

No. Everything here is one-dimensional: both bodies travel along the same line and direction reduces to a sign. A side impact or a glancing blow needs momentum resolved into two components and conserved in each separately. For a straight rear-end shunt, a head-on meeting or anything on a track, one dimension is the whole problem.

Where does the lost kinetic energy actually go?

Into permanent deformation first, then heat, then sound. Bending a steel panel takes work and the panel does not spring back, so that work is gone from the mechanical account. In the head-on example, 408,333.3 J disappears this way, which is the energy it would take to lift a one metric ton car 41.6 m straight up, and it is precisely what the crumple zone exists to spend.

Can I use pounds and mph and still trust the answer?

Yes. The factors are the defined ones rather than the rounded ones: 1 lb = 0.45359237 kg and 1 mph = 0.44704 m/s, both exact by international agreement. The conversion runs before the physics and the converted SI values are printed, so you can see the numbers the formula was actually given.

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