Conservation of Momentum
Calculate final velocities of two colliding objects using the law of conservation of momentum m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. Supports elastic and perfectly inelastic collisions with momentum and kinetic energy analysis and interactive charts.
About This Calculator
The Conservation of Momentum Calculator helps you solve two-body collision problems using the fundamental physics principle that the total momentum of an isolated system remains constant before and after a collision. This calculator is ideal for physics students, educators, engineers, and anyone studying mechanics and collision dynamics.
The law of conservation of momentum states that m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, where m is mass, u is initial velocity, and v is final velocity. The calculator supports two collision types: Perfectly elastic collisions where both momentum and kinetic energy are conserved (objects bounce off each other, like billiard balls), and Perfectly inelastic collisions where the objects stick together and move with a common velocity (like a bullet embedding in a target).
For elastic collisions, the calculator uses the simultaneous solution of momentum and kinetic energy conservation equations to determine both final velocities: v₁ = ((m₁ − m₂)u₁ + 2m₂u₂)/(m₁ + m₂) and v₂ = ((m₂ − m₁)u₂ + 2m₁u₁)/(m₁ + m₂). For perfectly inelastic collisions, the common final velocity is simply v = (m₁u₁ + m₂u₂)/(m₁ + m₂).
The results display final velocities of both objects, total momentum before and after (verifying conservation), kinetic energy before and after the collision, and the amount of kinetic energy lost or converted. Interactive bar charts compare momentum distribution between objects before and after collision, while pie charts visualize kinetic energy changes.
Regional Notes
The conservation of momentum is a universal physics law that applies worldwide — it is independent of region, currency, or measurement system. The calculator uses standard SI units (kg for mass, m/s for velocity, kg·m/s for momentum, and joules for kinetic energy) as used in physics education across India (CBSE/ICSE), the United States (AP Physics), the United Kingdom (A-Level Physics), and all other countries.
Frequently Asked Questions
What is the law of conservation of momentum?
The law of conservation of momentum states that the total momentum of an isolated system remains constant before and after a collision. Mathematically, m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, where m is mass, u is initial velocity, and v is final velocity of each object.
What is the difference between elastic and inelastic collisions?
In an elastic collision, both momentum and kinetic energy are conserved — objects bounce off each other without deformation. In a perfectly inelastic collision, objects stick together and move with a common final velocity; momentum is conserved but kinetic energy is partially converted to heat, sound, or deformation.
How do you calculate final velocities in an elastic collision?
For a perfectly elastic collision between two objects, the final velocities are v₁ = ((m₁ − m₂)u₁ + 2m₂u₂)/(m₁ + m₂) and v₂ = ((m₂ − m₁)u₂ + 2m₁u₁)/(m₁ + m₂). Both momentum and kinetic energy are conserved.
How do you calculate final velocity in a perfectly inelastic collision?
In a perfectly inelastic collision, the objects stick together and move with a common velocity v = (m₁u₁ + m₂u₂)/(m₁ + m₂). This formula comes from conserving momentum while setting both final velocities equal.
What are real-world examples of conservation of momentum?
Common examples include billiard ball collisions (nearly elastic), car crashes (inelastic), the recoil of a gun when firing a bullet, rocket propulsion where exhaust gases propel the rocket forward, and Newton's cradle demonstrating momentum transfer through a series of balls.
Is kinetic energy always conserved in a collision?
No, kinetic energy is only conserved in perfectly elastic collisions. In inelastic collisions, some kinetic energy is converted into other forms such as heat, sound, or plastic deformation. However, the total energy (including heat and deformation) is always conserved according to the law of conservation of energy.
What does it mean when momentum is conserved but kinetic energy is not?
When momentum is conserved but kinetic energy is not, the collision is inelastic. The missing kinetic energy has been transformed into other forms — typically heat from friction, sound waves from the impact, or work done deforming the objects. The total energy of the system including these forms remains constant.
Can momentum be negative in calculations?
Yes, momentum can be negative when the velocity is in the opposite direction to the chosen positive reference direction. In this calculator, enter velocities with appropriate signs — positive for one direction and negative for the opposite. The conservation equation handles signs automatically.