Momentum Calculator — Physics Solver

How Momentum Calculations Work

Momentum is one of the most fundamental quantities in physics, defined as the product of an object's mass and its velocity. The formula p = mv gives you the linear momentum in kilogram-metres per second (kg·m/s). Momentum is a vector quantity — it has both magnitude and direction, determined by the direction of velocity. A heavier object moving at the same speed as a lighter one carries more momentum, and a faster object carries more momentum than a slower one of equal mass.

You can rearrange the momentum formula to solve for unknown quantities. To find mass when momentum and velocity are known, use m = p/v. To find velocity from momentum and mass, use v = p/m. These rearrangements are essential when working through multi-step physics problems involving Newton's second law, which can also be expressed as F = dp/dt — force equals the rate of change of momentum.

Conservation of Momentum in Collisions

The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. This principle is the foundation for solving collision problems in physics. Before a collision, the total momentum of all objects equals the total momentum after the collision: m1·v1 + m2·v2 = m1·v1' + m2·v2'. This holds true regardless of whether the collision is elastic or inelastic.

Conservation of momentum applies to everything from subatomic particle interactions to car crashes and rocket propulsion. When a rocket expels exhaust gases backward, the gases gain momentum in one direction, and the rocket gains equal momentum in the opposite direction, keeping the total at zero. Understanding momentum conservation allows engineers to design safer vehicles, physicists to predict particle behavior, and sports scientists to analyze impact forces.

Elastic vs Inelastic Collisions

In an elastic collision, both momentum and kinetic energy are conserved. The objects bounce off each other without any energy lost to deformation, heat, or sound. Billiard ball collisions and atomic-level particle interactions closely approximate elastic collisions. The final velocities can be calculated using the formulas: v1' = ((m1−m2)v1 + 2m2·v2)/(m1+m2) and v2' = ((m2−m1)v2 + 2m1·v1)/(m1+m2).

In a perfectly inelastic collision, the objects stick together after impact and move as one combined mass. Momentum is still conserved, but kinetic energy is not — some energy converts to heat, sound, or deformation. The final velocity is v_f = (m1·v1 + m2·v2)/(m1+m2). The kinetic energy lost can be calculated by comparing the total kinetic energy before and after the collision. Most real-world collisions are inelastic to some degree.

Impulse and Momentum Change

Impulse is defined as the change in momentum of an object, calculated as J = F·t where F is the average force applied and t is the time duration. By Newton's second law, impulse equals the change in momentum: J = Δp = m·Δv. This relationship explains why extending the time of impact reduces the force — airbags and padding work by increasing the collision time, thereby reducing the peak force experienced by the body. Impulse is measured in newton-seconds (N·s), which is equivalent to kg·m/s, the same unit as momentum.

Momentum Calculator Worked Examples and Unit Conversions

Most wrong answers in a momentum problem are unit errors, not algebra errors. The formula p = mv only returns kg·m/s when mass is in kilograms and velocity in metres per second, so convert first, then multiply. Grams divide by 1000; tonnes multiply by 1000; pounds-mass multiply by 0.45359237. For velocity, km/h divide by 3.6, mph multiply by 0.44704, and feet per second multiply by 0.3048. Worked example: a 1500 kg car at 72 km/h is 72 ÷ 3.6 = 20 m/s, so p = 1500 × 20 = 30 000 kg·m/s. A 145 g baseball at 40 m/s is 0.145 × 40 = 5.8 kg·m/s. Direction is part of the answer — in one dimension, pick a positive axis and give any object moving the other way a negative velocity, because two equal masses closing head-on have a total momentum near zero even though neither is at rest. For the underlying definition and SI units, see the NIST SI units reference. Updated 2026-09-14.

Momentum vs Kinetic Energy: Why They Give Different Answers

Momentum (p = mv) and kinetic energy (KE = ½mv²) both grow with mass and speed, but they are not interchangeable and a problem that asks for one cannot be answered with the other. Momentum is a vector, so direction matters and opposite motions cancel; kinetic energy is a scalar and is always positive. Doubling speed doubles momentum but quadruples kinetic energy, which is why stopping distance grows so much faster than impact momentum suggests. The practical test: momentum is conserved in every collision, elastic or inelastic, while kinetic energy is only conserved in a perfectly elastic one. So use momentum to find velocities after a crash, and the energy difference to find how much was lost to heat and deformation. Momentum's SI unit is the kilogram metre per second (kg·m/s), built from the base units defined by NIST — SI Units.