Momentum Calculator

Momentum is a physical quantity that indicates how much motion an object in motion has. Any moving object has momentum and the greater the mass of an object and the higher its speed, the greater the momentum. This calculator calculates momentum from mass and velocity and can also be used to determine one of these quantities if the other two are known.

What is Momentum?

Momentum is a quantity that describes the motion of an object. It is the product of mass and velocity, and it is also a vector, meaning that it has both magnitude and direction. The direction of momentum is in the same direction as the motion of the object. If the direction of motion is opposite to the chosen positive direction, then the momentum will be negative.

The greater the momentum of an object, the harder it is to bring it to a halt because it has more inertia. A large ship moving slowly into port can have significantly more momentum than a sprinter running at full speed. The huge mass makes up for the slow velocity. To stop such a ship requires continuous gentle force over time.

The formula for momentum is:

Momentum is mass times velocity.

In this equation p is momentum, m is mass and v is velocity. If the mass doubles or the velocity doubles, then the momentum also doubles. By rearranging this same equation unknown values can be found.

Leave the desired numbers as placeholders and enter the other two numbers, then the calculation tool will rearrange the formula accordingly.

Unit of momentum

The international unit for momentum is kilogram meters per second (kg m/s). This is exactly the same as the unit for impulse, which is newton seconds (N s). That's because a change in momentum is an impulse. In older documents you may see gram centimeters per second (g cm/s) used, and in problems using the yard-pound system pound feet per second (lb ft/s) are sometimes used. You can switch the units for each field here, and the result will be converted instantly.

Example of calculating momentum:

The momentum of a baseball with a mass of 0.145 kg thrown at a speed of 40 m/s is...

The momentum of a car with mass 1500kg and speed 20m/s (roughly 72km/h) is...

This momentum is thousands of times larger than the momentum of a ball. This difference shows that even in a minor crash, a car can do much more damage than a fast thrown baseball.

Comparison of Momentum and Kinetic Energy

Both momentum and kinetic energy increase with the mass and speed of an object, but they are not the same thing. Momentum is the product of mass and velocity, so it increases proportionally to the velocity. Kinetic energy is half the product of mass and the square of the velocity, so it increases in proportion to the square of the velocity. If the velocity doubles, then momentum doubles while kinetic energy quadruples.

There is a simple relationship between them.

A heavy slow moving object can have the same momentum as a light fast moving object but their kinetic energy will be very different. Archers take advantage of this effect. A heavy arrow with moderate speed retains its momentum and penetrates deeper. A lighter faster projectile carries more energy but loses it quicker. This calculator shows both momentum and kinetic energy so you can compare them easily.

Conservation of momentum

In a closed system where no external forces are acting upon it, the total momentum remains constant. The amount of momentum an object has before a reaction is the same as the amount of momentum it has after the reaction. This explains why when a bullet is fired from a gun, the bullet goes forward and the gun recoils backward, and also why rockets go up when they expel hot gases downward. A change in momentum in one direction is balanced by an equal and opposite change in momentum in the other direction.

You can open the Collision option to see how two objects collide head on. If you select a perfectly inelastic collision, then the two objects will stick together and move off as one. You can also choose an elastic collision where the objects bounce apart and kinetic energy is conserved. The calculator will calculate the final velocity and show that total momentum has been conserved, as well as how much kinetic energy remains after the collision.

Impulse: How momentum changes

The momentum only changes when a force is applied over some time. The product of the force and the time is the impulse, which equals to the change in momentum.

A small force acting for a long time can produce the same change in momentum as a large force acting for a short time. This is the principle behind airbags, crumple zones and the motion of a catcher pulling his hand back when catching a ball. All these things lengthen the time over which the collision force acts, keeping it within manageable limits.

Relativistic momentum.

At everyday speeds the formula p = m * v applies, where 'p' is momentum, 'm' is mass and 'v' is velocity. In most practical cases this formula gives a sufficiently accurate answer. However, as an object approaches the speed of light, its increase in momentum exceeds what would be predicted by simply multiplying mass and velocity.

The full formula multiplies classical momentum by the Lorentz factor.

At a speed of one tenth the speed of light, the correction term is less than one percent. However, as speeds approach that of light, momentum increases without limit. This unbounded increase is why objects with mass can never reach the speed of light. By activating the relativistic option you will be able to display the Lorentz factor at high speeds and the corrected momentum.