Conservation of Momentum Calculator

Analyze one-dimensional collisions between two objects.

Select the collision type, enter the masses and initial velocities of the two objects, and click 'Calculate' to find the final velocities and energy changes.

Object 1

Object 2

Practical Examples

Load an example to see how the calculator works for different scenarios.

Inelastic Collision: Train Cars Coupling

inelastic

A 10,000 kg train car moving at 2 m/s collides and couples with a stationary 5,000 kg car.

Type: inelastic

m₁: 10000, u₁: 2

m₂: 5000, u₂: 0

Elastic Collision: Billiard Balls

elastic

A 0.17 kg cue ball moving at 5 m/s hits a stationary 0.16 kg object ball head-on.

Type: elastic

m₁: 0.17, u₁: 5

m₂: 0.16, u₂: 0

Elastic Collision: Equal Masses

elastic

Two objects of equal mass (2 kg) collide elastically. The first moves at 4 m/s, the second at -2 m/s.

Type: elastic

m₁: 2, u₁: 4

m₂: 2, u₂: -2

Inelastic Collision: Head-On

inelastic

A 1200 kg car moving at 20 m/s collides head-on with an 1800 kg truck moving at -15 m/s. They stick together.

Type: inelastic

m₁: 1200, u₁: 20

m₂: 1800, u₂: -15

Other Titles
Understanding the Conservation of Momentum: A Comprehensive Guide
Explore the fundamental principles of momentum, collisions, and how this calculator simplifies complex physics problems.

What is the Conservation of Momentum?

  • The Core Principle
  • Momentum as a Vector
  • Closed Systems
The law of conservation of momentum is a fundamental principle in physics, stating that for a closed system, the total momentum remains constant. A closed system is one that does not exchange any matter with its surroundings and is not acted on by external forces like friction. In simpler terms, if you add up the momentum of all objects in a system before they interact (e.g., collide), the total will be the same as the sum of their momentums after the interaction.
The Momentum Formula
Momentum (denoted by 'p') is the product of an object's mass (m) and its velocity (v). The formula is: p = m * v. Since velocity is a vector (having both magnitude and direction), momentum is also a vector. This means direction is crucial. In one-dimensional problems, we represent direction using positive and negative signs.

Types of Collisions: Elastic vs. Inelastic

  • Perfectly Inelastic Collisions
  • Perfectly Elastic Collisions
  • Real-World Collisions
Collisions are the most common application of momentum conservation. They are broadly classified into two types:
1. Inelastic Collisions
In an inelastic collision, kinetic energy is not conserved; it is transformed into other forms of energy like heat, sound, or potential energy during the deformation of the objects. A 'perfectly inelastic' collision is one where the maximum possible kinetic energy is lost, and the objects stick together after impact, moving with a single, common final velocity.
2. Elastic Collisions
In a perfectly elastic collision, both momentum and kinetic energy are conserved. The objects bounce off each other without any loss of kinetic energy. Collisions between subatomic particles or idealized billiard balls are often treated as elastic.

Step-by-Step Guide to Using the Calculator

  • Selecting Collision Type
  • Inputting Object Data
  • Interpreting the Results
This calculator simplifies the process into a few easy steps:
Step 1: Choose the 'Collision Type' from the dropdown menu ('Inelastic' or 'Elastic').
Step 2: Enter the mass and initial velocity for 'Object 1'. Ensure you use consistent units.
Step 3: Enter the mass and initial velocity for 'Object 2'. Remember that direction can be indicated with a negative sign.
Step 4: Click 'Calculate'. The results will appear below, showing final velocities, momentum, and kinetic energy values.

Real-World Applications

  • Vehicle Safety Engineering
  • Sports Science
  • Astrophysics and Rocket Science
The principle of momentum conservation is not just a theoretical concept; it's used to analyze and predict outcomes in many fields.
Examples
Car Crashes: Engineers use momentum conservation to design crumple zones in cars. By making the collision more inelastic, the impact time is extended, reducing the force experienced by passengers.
Billiards and Bowling: Players intuitively use momentum and energy transfer principles to predict how balls will interact.
Rocket Propulsion: A rocket expels gas at high velocity in one direction, causing the rocket to gain momentum in the opposite direction, propelling it forward.

Mathematical Derivation and Formulas

  • General Momentum Conservation Equation
  • Formula for Inelastic Collisions
  • Formulas for Elastic Collisions
The Foundation
The core equation for any collision in a closed system is: pinitial = pfinal or m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
Inelastic Collision Formula
Since the objects stick together, v₁ = v₂ = vfinal. The equation simplifies to m₁u₁ + m₂u₂ = (m₁ + m₂)vfinal. Solving for the final velocity gives: v_final = (m₁u₁ + m₂u₂) / (m₁ + m₂).
Elastic Collision Formulas
For elastic collisions, we also use the conservation of kinetic energy: ½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂². Solving these two conservation equations simultaneously yields the final velocities: v₁ = ((m₁ - m₂) / (m₁ + m₂))u₁ + (2m₂ / (m₁ + m₂))u₂ and v₂ = (2m₁ / (m₁ + m₂))u₁ + ((m₂ - m₁) / (m₁ + m₂))u₂.