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.