Momentum Calculator

What the momentum calculator does

Momentum is the quantity of motion an object carries, and it depends on both how much mass is moving and how fast. This calculator works it out from the mass and the velocity, in one, two, or three dimensions, and reports the momentum along each direction along with its overall size.

Below is what momentum means, the equation behind it, why it is treated as a vector, and a worked example.

How to use it

1. Enter the mass, in kilograms, grams, or pounds.
2. Choose the dimension you are working in, then enter the velocity along each axis.
3. Press Calculate for the momentum components and the total magnitude, or Reset to clear it.

What momentum is

Momentum is mass in motion. A thing has more of it the heavier it is and the faster it moves, so a slow-rolling truck and a fast-thrown ball can carry similar momentum despite their very different sizes. It is the property that makes a moving object hard to stop, and the heavier or faster it is, the harder.

Like velocity, momentum has a direction as well as a size, which makes it a vector. A ball moving east and the same ball moving west have momentum of equal size but opposite direction, and that direction matters when objects interact. This is why the calculator does not just give a single number but breaks the momentum down along each axis of motion.

The equation it uses

Momentum p is simply the mass m times the velocity v:

p = m v

Because velocity is a vector, momentum points the same way the object moves. When the motion has parts along more than one axis, each axis gets its own momentum, the mass times the velocity along that axis, and the overall size of the momentum is found by combining those parts.

Momentum in one, two, or three dimensions

In one dimension, motion is along a single line, and the momentum is just mass times that one speed, carrying a sign for which way along the line it goes. Most simple problems live here.

In two or three dimensions, the velocity has parts along each axis, and so does the momentum: the x-momentum is the mass times the x-velocity, and likewise for y and z. These components add as the sides of a right-angled box, so the total size of the momentum is the square root of the sum of the squares of the parts. The calculator reports each component separately, which is what you need when motion is not confined to a straight line, and then gives the overall magnitude.

Why momentum is so useful: it is conserved

The reason momentum earns its place in physics is a deep rule: in any interaction where no outside force intrudes, the total momentum before equals the total momentum after. It is conserved. Two objects can collide, push apart, or stick together, and however they rearrange their motion, their combined momentum is unchanged.

This is what lets you predict the outcome of a collision without knowing the messy details of the forces during it. It explains why a gun recoils backward as the bullet flies forward, why a rocket climbs by throwing exhaust down, and how skaters pushing off each other glide apart in balance. Add up the momentum as a vector before, and you know what it must add up to afterward.

Units and precision

Mass can be entered in kilograms, grams, or pounds, and velocity in metres per second, with the calculator converting mass to kilograms internally. Momentum is reported in kilogram-metres per second, the SI unit, for each component and for the total. Results are shown to a few decimal places, finer than the inputs usually warrant, so the precision is in the arithmetic rather than the measurement.

A worked example

Take a 2 kg object moving with a velocity of 3 m/s along the x-axis and 4 m/s along the y-axis.

The momentum along x is 2 × 3 = 6 kg·m/s, and along y it is 2 × 4 = 8 kg·m/s. The overall size is √(6² + 8²) = √100 = 10 kg·m/s. So the object carries 10 kg·m/s of momentum, directed along the combination of its two velocity parts.

Questions people ask

What is the formula for momentum?

Momentum equals mass times velocity, p = mv. Because velocity is a vector, momentum points in the direction the object moves.

What are the units of momentum?

The SI unit is the kilogram-metre per second (kg·m/s). It has no special name of its own.

Is momentum a vector?

Yes. It has both size and direction, the same direction as the velocity. That is why motion in two or three dimensions has a momentum component along each axis.

What does conservation of momentum mean?

It means that with no outside force, the total momentum of a system stays the same. The combined momentum before an interaction equals the combined momentum after, which is what makes collisions predictable.

References

A quick note on where the physics comes from. Momentum as mass times velocity, its nature as a vector, and its conservation are standard mechanics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The SI units follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 9.1, Linear Momentum and Force. https://openstax.org/books/university-physics-volume-1/pages/9-1-linear-momentum-and-force
2. HyperPhysics, Georgia State University, Momentum. http://hyperphysics.phy-astr.gsu.edu/hbase/mom.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811