Momentum Calculator finds momentum from mass and velocity, or the mass or velocity from a known momentum. It also totals several objects and shows kinetic energy and force to stop.
A velocity in the opposite direction gets a minus sign, and so does its momentum.
More Inputs and Units
Step-by-Step Working
- Known values in SI units: m = 1000 kg, v = 20 m/s
- Multiply: p = m × v = 1000 × 20 = 20,000 kg·m/s
- Kinetic energy: ½ × 1,000 × (20)² = 200,000 J
- Result: p = 20,000 kg·m/s
Momentum at Other Speeds
| Speed | Momentum | Kinetic Energy |
|---|---|---|
| 1/4 the speed | 5,000 kg·m/s | 12.5 kJ |
| 1/2 the speed | 10,000 kg·m/s | 50 kJ |
| Your speed | 20,000 kg·m/s | 200 kJ |
| 2 times the speed | 40,000 kg·m/s | 800 kJ |
| 4 times the speed | 80,000 kg·m/s | 3,200 kJ |
Momentum-Speed Graph
Momentum Calculator for Mass, Velocity and Totals
The Momentum Calculator has five settings under Calculate. The first multiplies a mass by a velocity. The next two rearrange the same formula to find the mass or the velocity. The last two handle a velocity given by its components and the total momentum of up to three objects.
It opens with a 1,000 kg car moving at 20 m/s, which has a momentum of 20,000 kg·m/s. Mass can be entered in kilograms, grams, pounds, ounces or grains, and velocity in m/s, km/h, ft/s or mph. A velocity in the opposite direction takes a minus sign, and so does its momentum.
Four cards follow. They give the kinetic energy, the force needed to stop the object in a chosen time, the momentum in other units, and the speed a person, a car and a baseball would each need to carry the same momentum. For the car, a 70 kg person would have to move at 285.71 m/s.
Momentum Formula
Momentum is mass times velocity, and its symbol is p. OpenStax College Physics defines it this way and notes that it points in the same direction as the velocity. This is the whole momentum calculation formula for one object.
$$p = m v$$
A 10 kg object moving at 2 m/s has a momentum of 20 kg·m/s. A 10 kg mass at rest has zero momentum, because its velocity is zero. The 1,000 kg car at 20 m/s has 20,000 kg·m/s, a thousand times the first example.
Dividing the momentum by either factor gives the other one, which is what the Mass and Velocity settings of the Momentum Calculator do. A momentum of 50,000 kg·m/s at 20 m/s belongs to a mass of 2,500 kg, and the same momentum carried by 1,000 kg means a velocity of 50 m/s.
$$m = \frac{p}{v}$$
$$v = \frac{p}{m}$$
In two or three dimensions, each part of the velocity has its own momentum. The size of the momentum is the mass times the speed, where the speed comes from the components. A 2 kg object moving at 3 m/s along x and 4 m/s along y has a speed of 5 m/s and a momentum of 10 kg·m/s.
$$p = m \sqrt{v_x^2 + v_y^2 + v_z^2}$$
Total Momentum Calculator
The formula for calculating total momentum adds the momentum of every object, each with its own sign. The Total of Several Objects setting takes two objects up front and a third under More Inputs.
$$p_{total} = m_1 v_1 + m_2 v_2 + m_3 v_3$$
Its default values are the OpenStax example of a football player and a football. A 110 kg player running at 8 m/s has 880 kg·m/s. A 0.41 kg ball thrown at 25 m/s has 10.25 kg·m/s, which the textbook rounds to 10.3. The player carries about 86 times the momentum of the ball.
Moving the same way, the two total 890.25 kg·m/s. If the ball flies toward the player, its velocity is −25 m/s and the total drops to 869.75 kg·m/s. In this setting the Momentum Calculator also reports the combined mass and the velocity of the center of mass, which is 8.06 m/s in the first case.
Total momentum matters because it stays the same in a collision. Finding the final momentum and velocities after two objects hit each other is the job of the Conservation of Momentum Calculator. A change in momentum caused by a force over time belongs to the Impulse and Momentum Calculator.
Momentum Calculator for a Bullet or Arrow
Shooters and archers quote weight in grains and speed in feet per second. The Momentum Calculator accepts both units directly. The table uses round example loads to show the range, with the momentum in metric and US units beside the kinetic energy.
| Projectile | Weight and speed | Momentum | In lb·ft/s | Kinetic energy |
|---|---|---|---|---|
| Small rimfire bullet | 40 gr at 1,200 ft/s | 0.948 kg·m/s | 6.86 | 127.9 ft·lbf |
| Arrow | 400 gr at 300 ft/s | 2.37 kg·m/s | 17.14 | 79.9 ft·lbf |
| Light pistol bullet | 115 gr at 1,150 ft/s | 2.612 kg·m/s | 18.89 | 337.6 ft·lbf |
| Light rifle bullet | 55 gr at 3,200 ft/s | 3.476 kg·m/s | 25.14 | 1,250.3 ft·lbf |
| Heavy pistol bullet | 230 gr at 850 ft/s | 3.861 kg·m/s | 27.93 | 368.9 ft·lbf |
| Medium rifle bullet | 150 gr at 2,800 ft/s | 8.295 kg·m/s | 60 | 2,610.8 ft·lbf |
Momentum and kinetic energy rank these differently. The slow, heavy pistol bullet has more momentum than the fast, light rifle bullet, yet less than a third of its energy. Energy grows with the square of the speed, and momentum grows only in step with it.
The arrow shows the same thing from the other side. It carries more momentum than the rimfire bullet with less energy. Arrow speed for a given bow comes from the Arrow Speed Calculator, and how well a bullet keeps its speed downrange is described by the Ballistic Coefficient Calculator.
Momentum, Force and Newton’s Laws
Is momentum equal to force?
Momentum is not a force. Force is the rate at which momentum changes. Stopping the 20,000 kg·m/s car in 4 s takes an average force of 5,000 N. Stopping it in 2 s takes 10,000 N. The second card of the Momentum Calculator works this out for any stopping time, with the stopping distance and the deceleration.
What is Newton’s second law in terms of momentum?
Newton stated his second law with momentum. The net force on an object equals its change in momentum divided by the time the change takes. When the mass is constant, this becomes the familiar force equals mass times acceleration.
$$F = \frac{\Delta p}{\Delta t}$$
What are the laws of momentum?
There are two ideas that textbooks call the rule for momentum. The first is the second law above, which says only a net force can change momentum. The second is conservation, which says the total momentum of a system stays constant when no outside force acts on it.
Newton’s first law is the special case with no force at all. An object keeps the momentum it has, at rest or moving in a straight line, until a net force acts. This is also called the law of inertia, since inertia is the tendency of a mass to resist changes in its motion.
Momentum Units and Symbol
The SI unit of momentum is the kilogram-meter per second, kg·m/s. It is the same as the newton-second. Mass goes in as kilograms, not grams, when the answer is wanted in SI units, and the Momentum Calculator converts grams, pounds and grains to kilograms before it multiplies.
The third card of the Momentum Calculator lists the result in other units. The car’s 20,000 kg·m/s is 144,660.3 lb·ft/s, or 4,496.2 lbf·s, which is the same as slug-feet per second. Result Units switches the headline between metric and US units.
The quantity is properly called linear momentum, to separate it from the angular momentum of something that spins. Its symbol p is the usual short form. At speeds near that of light the real formula gains a factor, written γmv, but at a tenth of light speed the correction is still only 0.5%.