Redshift Academy

Wolfram Alpha:

Search by keyword:

Astronomy

-

Celestial Coordinates

-

Celestial Navigation

-

-

Location of North and South Celestial Poles

Chemistry

-

Avogadro's Number

-

Balancing Chemical Equations

-

-

The Periodic Table

Classical Physics

-

Archimedes Principle

-

Bernoulli Principle

-

Blackbody (Cavity) Radiation and Planck's Hypothesis

-

Center of Mass Frame

-

Comparison Between Gravitation and Electrostatics

-

-

Coriolis Effect

-

Cyclotron Resonance

-

-

-

Double Slit Experiment

-

Elastic and Inelastic Collisions

-

Electric Fields

-

-

-

-

Gauss's Law of Universal Gravity

-

Gravity - Force and Acceleration

-

-

Ideal and Non-Ideal Gas Laws (van der Waal)

-

-

-

-

-

-

Kinetic Theory of Gases

-

Kirchoff's Laws

-

Laplace's and Poisson's Equations

-

Lorentz Force Law

-

Maxwell's Equations

-

Moments and Torque

-

-

One Dimensional Wave Equation

-

Pascal's Principle

-

Phase and Group Velocity

-

Planck Radiation Law

-

Poiseuille's Law

-

Radioactive Decay

-

Refractive Index

-

Rotational Dynamics

-

Simple Harmonic Motion

-

Specific Heat, Latent Heat and Calorimetry

-

Stefan-Boltzmann Law

-

-

The Laws of Thermodynamics

-

The Zeeman Effect

-

Wien's Displacement Law

-

Young's Modulus

Climate Change

-

Cosmology

-

Penrose Diagrams

-

-

Cosmic Background Radiation and Decoupling

-

-

-

Friedmann-Robertson-Walker Equations

-

Geometries of the Universe

-

-

Inflation Theory

-

Introduction to Black Holes

-

Olbers' Paradox

-

-

Stephen Hawking's Last Paper

-

Stephen Hawking's PhD Thesis

-

The Big Bang Model

Finance and Accounting

-

-

-

Brownian Model of Financial Markets

-

Capital Structure

-

Dividend Discount Formula

-

Lecture Notes on International Financial Management

-

-

Periodically and Continuously Compounded Interest

-

Repurchase versus Dividend Analysis

General Relativity

-

Accelerated Reference Frames - Rindler Coordinates

-

Catalog of Spacetimes

-

Curvature and Parallel Transport

-

Dirac Equation in Curved Spacetime

-

Einstein's Field Equations

-

-

Gravitational Time Dilation

-

Gravitational Waves

-

-

Quantum Gravity

-

Relativistic, Cosmological and Gravitational Redshift

-

Ricci Decomposition

-

-

Stress-Energy Tensor

-

Stress-Energy-Momentum Tensor

-

-

The Area Metric

-

The Equivalence Principal

-

The Essential Mathematics of General Relativity

-

The Induced Metric

-

The Metric Tensor

-

Vierbein (Frame) Fields

-

World Lines Refresher

Lagrangian and Hamiltonian Mechanics

-

Classical Field Theory

-

Euler-Lagrange Equation

-

Ex: Newtonian, Lagrangian and Hamiltonian Mechanics

-

Hamiltonian Formulation

-

Liouville's Theorem

-

Symmetry and Conservation Laws - Noether's Theorem

Macroeconomics

-

Lecture Notes on International Economics

-

Lecture Notes on Macroeconomics

-

Macroeconomic Policy

Mathematics

-

Amplitude, Period and Phase

-

Arithmetic and Geometric Sequences and Series

-

-

Augmented Matrices and Cramer's Rule

-

Basic Group Theory

-

Basic Representation Theory

-

Binomial Theorem (Pascal's Triangle)

-

Building Groups From Other Groups

-

Completing the Square

-

Complex Numbers

-

Composite Functions

-

Conformal Transformations

-

Conjugate Pair Theorem

-

Contravariant and Covariant Components of a Vector

-

Derivatives of Inverse Functions

-

Double Angle Formulas

-

Eigenvectors and Eigenvalues

-

Euler Formula for Polyhedrons

-

Factoring of a³ +/- b³

-

Fourier Series and Transforms

-

-

Gauss's Divergence Theorem

-

Grassmann and Clifford Algebras

-

Heron's Formula

-

Index Notation (Tensors and Matrices)

-

-

Integration By Parts

-

Introduction to Conformal Field Theory

-

Inverse of a Function

-

Law of Sines and Cosines

-

Line Integrals, ∮

-

Logarithms and Logarithmic Equations

-

Matrices and Determinants

-

Matrix Exponential

-

Mean Value and Rolle's Theorem

-

Modulus Equations

-

Orthogonal Curvilinear Coordinates

-

Parabolas, Ellipses and Hyperbolas

-

Piecewise Functions

-

Polar Coordinates

-

Polynomial Division

-

-

-

Regular Polygons

-

-

Sets, Groups, Modules, Rings and Vector Spaces

-

Similar Matrices and Diagonalization

-

Spherical Trigonometry

-

Stirling's Approximation

-

Sum and Differences of Squares and Cubes

-

-

Symmetric Groups

-

Tangent and Normal Line

-

Taylor and Maclaurin Series

-

The Essential Mathematics of Lie Groups

-

The Integers Modulo n Under + and x

-

The Limit Definition of the Exponential Function

-

Tic-Tac-Toe Factoring

-

Trapezoidal Rule

-

-

Vector Calculus

-

Volume Integrals

Microeconomics

-

Marginal Revenue and Cost

Particle Physics

-

Feynman Diagrams and Loops

-

Field Dimensions

-

Helicity and Chirality

-

Klein-Gordon and Dirac Equations

-

Regularization and Renormalization

-

Scattering - Mandelstam Variables

-

Spin 1 Eigenvectors

-

The Vacuum Catastrophe

Probability and Statistics

-

Box and Whisker Plots

-

Categorical Data - Crosstabs

-

Chebyshev's Theorem

-

Chi Squared Goodness of Fit

-

Conditional Probability

-

Confidence Intervals

-

-

-

Factor Analysis

-

Hypothesis Testing

-

Linear Regression

-

Monte Carlo Methods

-

Non Parametric Tests

-

-

Pearson Correlation

-

Permutations and Combinations

-

Pooled Variance and Standard Error

-

Probability Distributions

-

Probability Rules

-

Sample Size Determination

-

Sampling Distributions

-

Set Theory - Venn Diagrams

-

Stacked and Unstacked Data

-

Stem Plots, Histograms and Ogives

-

Survey Data - Likert Item and Scale

-

-

Programming and Computer Science

-

-

How this site works ...

-

More Programming Topics

-

MVC Architecture

-

Open Systems Interconnection (OSI) Standard - TCP/IP Protocol

-

Public Key Encryption

Quantum Field Theory

-

Creation and Annihilation Operators

-

Field Operators for Bosons and Fermions

-

Lagrangians in Quantum Field Theory

-

Path Integral Formulation

-

Relativistic Quantum Field Theory

Quantum Mechanics

-

Basic Relationships

-

-

-

Clebsch-Gordan Coefficients

-

-

-

Electron Orbital Angular Momentum and Spin

-

Entangled States

-

Heisenberg Uncertainty Principle

-

Ladder Operators

-

Multi Electron Wavefunctions

-

Pauli Exclusion Principle

-

Pauli Spin Matrices

-

Photoelectric Effect

-

Position and Momentum States

-

Probability Current

-

Schrodinger Equation for Hydrogen Atom

-

Schrodinger Wave Equation

-

Schrodinger Wave Equation (continued)

-

Spin 1/2 Eigenvectors

-

The Differential Operator

-

The Essential Mathematics of Quantum Mechanics

-

The Observer Effect

-

-

The Schrodinger, Heisenberg and Dirac Pictures

-

The WKB Approximation

-

Time Dependent Perturbation Theory

-

Time Evolution and Symmetry Operations

-

Time Independent Perturbation Theory

-

Semiconductor Reliability

-

The Weibull Distribution

Solid State Electronics

-

Band Theory of Solids

-

Fermi-Dirac Statistics

-

Intrinsic and Extrinsic Semiconductors

-

-

The P-N Junction

Special Relativity

-

-

Electromagnetic 4 - Potential

-

Energy and Momentum, E = mc²

-

Lorentz Invariance

-

Lorentz Transform

-

Lorentz Transformation of the EM Field

-

Newton versus Einstein

-

Spinors - Part 1

-

Spinors - Part 2

-

The Lorentz Group

-

Velocity Addition

Statistical Mechanics

-

Black Body Radiation

-

Entropy and the Partition Function

-

The Harmonic Oscillator

-

String Theory

-

Bosonic Strings

-

Extra Dimensions

-

Introduction to String Theory

-

Kaluza-Klein Compactification of Closed Strings

-

Strings in Curved Spacetime

-

Toroidal Compactification

Superconductivity

-

-

Introduction to Superconductors

-

Superconductivity (Lectures 1 - 10)

-

Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

-

Chiral Superfields

-

Generators of a Supergroup

-

Grassmann Numbers

-

Introduction to Supersymmetry

-

The Gauge Hierarchy Problem

test

-

The Standard Model

-

Electroweak Unification (Glashow-Weinberg-Salam)

-

Gauge Theories (Yang-Mills)

-

Gravitational Force and the Planck Scale

-

Introduction to the Standard Model

-

Isospin, Hypercharge, Weak Isospin and Weak Hypercharge

-

Quantum Flavordynamics and Quantum Chromodynamics

-

Special Unitary Groups and the Standard Model - Part 1

-

Special Unitary Groups and the Standard Model - Part 2

-

Special Unitary Groups and the Standard Model - Part 3

-

Standard Model Lagrangian

-

The Higgs Mechanism

-

The Nature of the Weak Interaction

Topology

-

Units, Constants and Useful Formulas

-

-

Last modified: January 26, 2018


Center of Mass Frame
--------------------

We can always choose a frame of reference where the center of 
mass is stationary.  Under these circumstances Σp = 0.  Consider, 
the following inelastic collision between 2 masses moving in the 
same direction.

Before:

->v₁       ->v₂  
m₁         m₂   

After (masses stick together):

->v₃
m₁ + m₂

In the laboratory frame:

Momentum:  m₁v₁ + m₂v₂ = (m₁ + m₂)v₃

Energy:  (1/2)m₁v₁² + (1/2)m₂v₂² = (1/2)(m₁ + m₂)v₃²

In the COM frame:

Momentum:  m₁(v₁ - v₃) + m₂(v₂ - v₃) = 0

Energy:  (1/2)m₁(v₁ - v₃)² + (1/2)m₂(v₂ - v₃)² = (1/2)(m₁ + m₂)v₃²

Add some numbers

m₁ = 1, v₁ = 4
m₂ = 2, v₂ = 2

Laboratory frame:

Momentum: 1*4 + 2*2 = 3*v₃  

          => v₃ = 8/3 = 2.67

Energy: (1/2)*1*16 + (1/2)*2*4 ≠ (1/2)*3*64/9 

         => 1.35 units of energy lost in collision.

COM frame:

Momentum: 1*(4 - 2.67) + 2*(2 - 2.67) 

          => 0

Energy: (1/2)*1*(4 - 2.67)² + (1/2)*2*(2 - 2.67)² 

         => 1.35 units of energy lost in collision.