Momentum Calculator
Solve p = mv for momentum, mass or velocity, with kinetic energy and a relativistic validity check.
🎱 What is Momentum?
Momentum is the quantity of motion an object carries. In classical mechanics the linear momentum of a body is its mass multiplied by its velocity, p = m v. A heavy truck rolling slowly and a light bullet flying fast can have momenta of similar size, because the product of mass and speed is what counts. Momentum is a vector, so it points in the same direction as the velocity, and its SI unit is the kilogram metre per second (kg m/s), which is the same as the newton second (N s).
Momentum matters because it is tied directly to force. Newton's second law can be written as force equals the rate of change of momentum, and the impulse delivered by a force over a time equals the change in momentum. That is why a car airbag, a catcher's glove and a crumple zone all work by stretching the time over which momentum changes, which lowers the average force. Momentum is also conserved: in an isolated system with no external force, the total momentum before a collision equals the total afterwards. This makes it the key tool for analysing collisions, rocket thrust, recoil and explosions.
This calculator solves p = m v in whichever direction you need. Find the momentum of a moving object from its mass and speed, find the mass of an object from its momentum and speed, or find the speed from the momentum and mass. It accepts kilograms, grams, pounds and tonnes, and metres per second, kilometres per hour, miles per hour and feet per second, then reports everything in SI units with lb ft/s as an extra. It also gives the kinetic energy computed from momentum, KE = p²/(2m).
One common misconception is that momentum and kinetic energy are the same thing. They are not: doubling the speed doubles the momentum but quadruples the kinetic energy. Another is that p = m v is always exact. It is an excellent approximation at everyday speeds, but at speeds that are a sizeable fraction of the speed of light the true momentum is γ m v, where γ is the Lorentz factor. The validity check in the results shows exactly how large that correction is for your inputs.
📐 Momentum Formula
The relativistic check reports (γ − 1) × 100, the percent by which γ m v exceeds m v. It is computed with a numerically stable form so that tiny corrections at everyday speeds are not lost to rounding. For sources, see the Wikipedia article on momentum and OpenStax College Physics, section 8.1.
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A 1500 kg car at 25 m/s
Family car on a city road, with a 70 kg person as the comparison mass
Example 2 — A 0.145 kg baseball at 40 m/s
A fastball of about 144 km/h
Example 3 — Mass of a runner from momentum
A runner with 560 kg m/s of momentum at 8 m/s
Example 4 — Speed of a 100 g tennis ball
A tennis ball (100 g) carrying 6 kg m/s
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for momentum?
Linear momentum is p = m v, mass times velocity. With mass in kilograms and velocity in metres per second, momentum comes out in kg m/s. A 1500 kg car moving at 25 m/s has p = 1500 x 25 = 37,500 kg m/s. Momentum is a vector that points the same way as the velocity.
What are the units of momentum?
The SI unit is the kilogram metre per second (kg m/s). It is identical to the newton second (N s) because 1 N = 1 kg m/s^2. In US customary units momentum is often given in lb ft/s, where 1 lb ft/s equals about 0.138255 kg m/s.
How do I find mass from momentum and velocity?
Rearrange p = m v to m = p / v. Choose the Find mass tab, enter the momentum and the speed, and the calculator divides them. A runner with 560 kg m/s of momentum at 8 m/s has a mass of 560 / 8 = 70 kg.
How do I find velocity from momentum and mass?
Use v = p / m. Choose the Find velocity tab and enter the momentum and the mass. A tennis ball of mass 0.1 kg with 6 kg m/s of momentum is moving at 6 / 0.1 = 60 m/s, which is 216 km/h.
How is kinetic energy related to momentum?
Kinetic energy is KE = p^2 / (2m), which equals 1/2 m v^2. For the 1500 kg car at 25 m/s, p = 37,500 kg m/s and KE = 37,500^2 / 3000 = 468,750 J. At the same momentum, a lighter object always has more kinetic energy.
Is momentum the same as force or impulse?
No. Momentum is mass times velocity. Force is the rate of change of momentum, F = dp/dt. Impulse is force times the time it acts, J = F x t, and it equals the change in momentum. Impulse and momentum share the units kg m/s.
When does p = mv stop being accurate?
The classical formula underestimates the true relativistic momentum gamma m v by (gamma - 1) x 100 percent. That is 0.005 percent at 1 percent of the speed of light, about 0.5 percent at 10 percent of c, and about 15.5 percent at half of c. Everyday speeds are far below these.
Can momentum be negative?
In one dimension, yes. Choosing a positive direction means an object moving the opposite way has negative momentum. This calculator takes speeds as positive magnitudes, so reverse the sign yourself if the motion is in the negative direction.