Momentum Calculator

What momentum measures

Momentum is mass in motion. Every moving object carries it, and the amount is the object's mass multiplied by its velocity. A shopping cart rolling at walking speed has a little; a loaded truck at highway speed has a lot. Physicists write the quantity as p and the relation as p = m x v. The letter p is just the symbol chosen long ago, and the unit that falls out of the multiplication is kilogram meters per second, written kg m/s.

Momentum earns its own name because it is the quantity that forces change and that collisions conserve. Push on an object and its momentum changes by exactly the force times how long you pushed. Let two objects collide with nothing else interfering, and the total momentum before the hit equals the total after. Those two rules cover car crashes, rocket thrust and billiards, and OpenStax College Physics builds its whole collision chapter on them.

The one formula and its units

The whole calculator runs on one line: p = m x v. Mass in kilograms, velocity in meters per second, momentum in kilogram meters per second. Direction counts, because momentum is a vector: a cart moving east at 3 m/s and one moving west at 3 m/s carry momenta of +3 and -3 kg m/s. In one dimensional problems pick a positive direction and give every velocity its sign before multiplying.

Conversions come up as soon as a problem quotes pounds or miles per hour. Pounds to kilograms divides by 2.205, and miles per hour to meters per second multiplies by 0.447. Running the other way, 1 kg m/s equals 7.233 pound feet per second, the unit many United States engineering handbooks print. NASA's Glenn Research Center keeps a short page on momentum that works the same definitions in both unit systems.

The same momentum reads differently in other unit systems.

Worked examples at everyday scale

The table runs p = m x v across six objects people meet every day. The last column multiplies each mass by half the square of its speed for comparison in the next section.

The car row in full: 1,500 x 25 = 37,500 kg m/s. The bullet row holds the surprise: leaving the muzzle at 900 m/s, a 10 gram bullet carries 9 kg m/s, barely more than the pitched baseball and less than the jogger. Mass beats speed in the momentum column, which is why a slow barge can crush a dock fender that a fast rowboat would only nudge.

Momentum against kinetic energy

Both numbers grow with speed and both come from the same two inputs, but they answer different questions. Kinetic energy, 1/2 m v^2, measures the work needed to bring the object to a stop. Momentum, m v, measures the quantity a force transfers over time. Double the speed and momentum doubles while energy quadruples, which the table shows: the freight car runs 8 times the car's momentum but 3.2 times its energy at half the speed and twenty times the mass.

Crash engineering lives on the split. Barrier design counts joules to absorb; airbag timing counts kg m/s to cancel. When both numbers are known there is a shortcut linking them: kinetic energy equals momentum squared divided by twice the mass, KE = p^2 / (2m). Check it with the runner: 280 squared divided by 140 gives 560 joules, matching the table.

Impulse: how force changes momentum

A force held for a time changes momentum by the product of the two. That product is called impulse: J = F x Δt = Δp. The line explains why airbags exist. A 70 kg occupant at 14 m/s carries 980 kg m/s that must reach zero. Against a hard dashboard that takes about 0.01 seconds, an average force of 98,000 newtons. An airbag stretches the stop to about 0.10 seconds, and the force drops to 9,800 newtons. Same change in momentum, one tenth the force, because the time grew tenfold.

Sports run the same arithmetic in reverse. A pitcher extends the contact time of hand on ball to build a big momentum change with a modest force. A boxer rides a punch by moving the head backward, stretching the stopping time to cut the peak load on the skull. In both directions the product F x Δt stays the boss; force and time trade off against each other.

Conservation of momentum in collisions

When two objects collide and nothing external pushes on the pair, the total momentum before the hit equals the total after. For two masses in one dimension: m1 u1 + m2 u2 = m1 v1 + m2 v2, with u the velocities before and v the velocities after. If the objects stick together, the collision is called perfectly inelastic, and the shared final velocity falls out of one division.

Take a 1,000 kg car sliding at 20 m/s into a stationary 1,500 kg car, bumpers locking. The combined 2,500 kg rolls on at (1,000 x 20) / 2,500 = 8 m/s. Momentum vanished nowhere; the missing speed went into crumpling metal and heat. In an elastic collision, billiard ball style, the objects bounce apart and kinetic energy survives the hit as well as momentum. Real crashes land between the two cases, which is why safety ratings measure energy absorbed rather than trusting either ideal.

The locking collision from the takeaways, step by step.

Slips that skew the number

• Miles per hour into a meters per second formula. The factor between them is 2.237, so skipping the conversion lands the answer 2.237 times high. Convert first, multiply second.
• Dropping the sign. Head on momenta subtract. Adding 3 and -3 as 6 instead of 0 turns a dead stop into a doubling.
• Grams entered as kilograms. A 10 gram bullet is 0.010 kg. Typing 10 makes the momentum a thousand times too big and the kinetic energy a million times too big.
• Reading kg m/s as energy. The units do not convert. 37,500 kg m/s is not 37,500 joules, and the two numbers answer different questions.
• Rounding the mass early. Cutting 0.145 kg to 0.15 shifts a 40 m/s baseball from 5.8 to 6.0 kg m/s, a 3 percent error from one careless step.

One habit prevents most of this: write the units next to every number and convert before the first multiplication, not after.

Another slip is mixing up speed directions. Momentum carries a sign along an axis: a 900 kg car driving east at 20 m/s has +18,000 kg m/s while a 900 kg car driving west at 20 m/s has -18,000 kg m/s. Head on, the pair sums to zero, which is exactly why equal masses moving at equal speeds in opposite directions stop dead. The calculator treats every entry as a magnitude, so give it speeds, not signed velocities, unless you are doing the collision bookkeeping by hand.

Neighboring tools for the rest of the motion

Motion pages on this site share the same inputs. The force calculator covers F = m a, the impulse calculator works the force and time side of the momentum change, and the kinetic energy calculator handles the energy column of the table above. For the ingredients of p = m v, the mass calculator and the acceleration calculator fill in the pieces, the terminal velocity calculator finds the speed where drag balances weight, and the centripetal force calculator takes the curved path cases.