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PF Insights Blog: Physics, Math and Science Articles

Physics, Math and Science, Explained

Velocity Time Function

[Velocity Time Function]

Revisiting the Velocity-Time Function

May 24, 2026/1 Comment/in Mechanics, Physics Tutorials/by Neil Parker

Kinematics: The Velocity-Time Function Author’s Preface: A Collaborative Approach to Lesson Development The intention behind what I hope will be a series of articles on Kinematics (this one being the first) is to begin an open, collaborative process of lesson development on Physics Forums. In order to expedite this workflow, I am making deliberate use…

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Recent Entries

ai problem solving

[ai problem solving]

AI Enriched Problem Solving

February 28, 2026

/

Introduction This article is written by way of a reference for my longstanding PF colleague and prolific poster of challenging Maths/geometry problems

Remote Operated Gate Control System

February 28, 2026

/

Introduction This project began life in 2017 when the author had just begun experimentation with GPIO pins on a Raspberry Pi 3. At that point in time

Inside the Scientific Box: History and Challenges Today

October 13, 2025

/

In Memoriam In Memory of Dr. Thomas J. LeCompte (1964-2025), Detector Designer and Champion of Education and Science. Prologue Defining "the box" Someone

Entangled Photon Polarization Qubits

[Entangled Photon Polarization Qubits]

Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality per Quantum Information Theory

September 29, 2025

/

In her YouTube video Bell's Theorem Experiments on Entangled Photons, Dr. Fugate shows how polarization-entangled photons violate Bell's inequality. In

Relativator (Circular Slide-Rule) – Simulated with Desmos

September 2, 2025

/

The Relativator (revisited) This is an update of my 2006 post (reconstructed in 2014) Relativator: The circular slide-rule for physicists. This is a

[dirac function]

What Exactly is Dirac’s Delta Function?

August 29, 2025

/

Introduction: "Convenient Notation"In Dirac's Principles of Quantum Mechanics published in 1930 he introduced a "convenient notation" he referred

quantum entanglement

[quantum entanglement]

Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

August 27, 2025

/

Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and

complex numbers

[complex numbers]

Fixing Things Which Can Go Wrong With Complex Numbers

July 20, 2025

/

Abstract This article will build on the hints about treating the complex numbers as a branched surface, briefly described and pictured in section 4.2

Fermat's Last Theorem

[Fermat's Last Theorem]

Fermat’s Last Theorem

May 18, 2025

/

Abstract Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states

[vector spaces]

Vector Spaces: Concepts, History, and Applications Guide

March 13, 2025

/

The Concept A vector space is an additively written abelian group together with a field that operates on it.Intuitive picture vs abstract definition Vector

Groups, The Path from a Simple Concept to Mysterious Results

February 2, 2025

/

Introduction The concept of a group is as simple as it gets: a set with a binary operation like addition and a couple of natural laws like the requirement

digital audio guide

[digital audio guide]

Hybrid AI Video Codecs and Modern Streaming Techniques

January 24, 2025

/

Introduction AI and Hybrid Codecs Modern Video codecs are all based on similar principles. Recently, these have been complemented by AI techniques,

[no tidal bulge]

Why the Tidal Bulge Doesn’t Exist – Tidal Theory Explained

December 31, 2024

/

Introduction Overview That there is no tidal bulge is the key premise of this article. Upper-level oceanography undergraduates and above know this. Yet

The Many Faces of Topology

December 6, 2024

/

Abstract Topology as a branch of mathematics is a bracket that encompasses many different parts of mathematics. It is sometimes even difficult to see

brownian motion

[brownian motion]

Brownian Motions and Quantifying Randomness in Physical Systems

August 26, 2024

/

Stochastic calculus has come a long way since Robert Brown described the motion of pollen through a microscope in 1827. It's now a key player in data science,

physics of reality

[physics of reality]

Quantum Reconstruction & Bell Correlations Explained

July 22, 2024

/

Comment on PBS Space Time video PBS Space Time produces some very good videos on the foundations of quantum mechanics (QM), so let me comment on their

Aspects Behind the Concept of Dimension in Various Fields

June 21, 2024

/

Abstract It took until the last century for physicists and mathematicians in the Netherlands to question the Euclidean concept of dimension as length,

complex numbers views

[complex numbers views]

Views On Complex Numbers

June 7, 2024

/

Abstract Why do we need yet another article about complex numbers? This is a valid question and I have asked it myself. I could mention that I wanted

Addition of Velocities (Velocity Composition) in Special Relativity

May 20, 2024

/

The "Addition of Velocities" formula (more correctly, the "Composition of Velocities" formula) in Special Relativity[tex]\frac{v_{AC}}{c}=\frac{ \frac{v_{AB}}{c}+\frac{v_{BC}}{c}

Schrodinger's cat

[Schrodinger's cat]

Schrdinger’s Cat and the Qbit

April 2, 2024

/

The concept of quantum superposition (or superposition for short) is very counterintuitive, as Schr##\ddot{\text{o}}##dinger noted in 1935 writing [1],

slinky drop experiment

[slinky drop experiment]

The Slinky Drop Experiment Analysed

March 28, 2024

/

The slinky drop is a rather simple experiment. In its most basic form, it requires only a popular toy for children, a stable hand, and a keen eye.

infinite universe

[infinite universe]

Open, Flat, or Closed Universe: What Curvature Tells Us

March 16, 2024

/

Direct answer: Cosmologists classify the universe as open, flat, or closed based on spatial curvature. Current cosmic microwave background (CMB) measurements

einstein field equations

[einstein field equations]

Einstein Field Equations Explained: Structure, Solutions, Facts

February 26, 2024

/

The Einstein Field Equations (EFE) are ten coupled, nonlinear differential equations that describe how matter and energy determine the curvature of spacetime.

Multi-Atwood Machine Assembly

[Multi-Atwood Machine Assembly]

How to Solve a Multi-Atwood Machine Assembly

February 16, 2024

/

IntroductionThe figure on the right shows a "double-double" Atwood machine with three ideal pulleys and four masses. All pulleys are released from

Lambert W Function in Finance

[Lambert W Function in Finance]

The Lambert W Function in Finance

February 9, 2024

/

Preamble The classical mathematician practically by instinct views the continuous process as the "real" process, and the discrete process as an approximation

Why Division by Zero Is Impossible: 10 Mathematical Reasons

January 11, 2024

/

A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs:What if it was

digital audio guide

[digital audio guide]

Sinc Function & Upsampling in Digital Audio Explained

December 14, 2023

/

Introduction This elaborates some of the claims in my insights article on digital audio. The Sinc Function Upsampling and Zero Insertion The first link

digital audio intro

[digital audio intro]

Modern DACs, Dither, DXD & High-Res Audio Explained

December 11, 2023

/

Introduction First, we need some background in Digital Signals. This can be mathematically quite advanced, but since I would like this article to be accessible

geometric series

[geometric series]

Series in Mathematics: From Zeno to Quantum Theory

November 7, 2023

/

Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century

Epsilontic limits and continuity

[Epsilontic limits and continuity]

Epsilontic – Limits and Continuity

October 6, 2023

/

Abstract I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I

The Poor Man’s Milli-Ohm Meter

September 1, 2023

/

Introduction In a previous article on measuring battery internal resistance, a simple technique for low-resistance measurement was outlined. In this article,

Differential Equation Systems and Nature

[Differential Equation Systems and Nature]

Differential Equation Systems and Nature

August 27, 2023

/

Abstract "Mathematics is the native language of nature." is a phrase that is often used when it comes to explaining why mathematics is all around in natural

Beginners Guide to Precalculus, Calculus and Infinitesimals

August 18, 2023

/

Introduction I am convinced students learn Calculus far too late. In my view, there has never been a good reason for this.In the US, they go through

Variable Mass Systems

[Variable Mass Systems]

How to Apply Newton’s Second Law to Variable Mass Systems

August 15, 2023

/

Introduction The applicability of Newton's second law in the oft-quoted "general form" $$\begin{align}\frac{d\mathbf{P}}{dt}=\mathbf{F}_{\text{ext}}\end{align}$$

what are numbers

[what are numbers]

What Are Numbers?

August 14, 2023

/

Introduction When doing mathematics, we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive. Going

world of algebras

[world of algebras]

Introduction to the World of Algebras

July 8, 2023

/

Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois

chatgpt-reliable

[chatgpt-reliable]

Why ChatGPT Is Unreliable: Design Limits and Misconceptions

July 5, 2023

/

ChatGPT's reliability I'll start with the simple fact: ChatGPT is not a reliable answerer to questions. Wolfram's explanation To try to explain why from

[Infinitesimals]

What Are Infinitesimals – Simple Version

June 4, 2023

/

Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say

quantum information

[quantum information]

How Quantum Information Theory Solves “the only mystery” of Quantum Mechanics

June 3, 2023

/

In Chapter 37 of "The Feynman Lectures on Physics Volume 1," Richard Feynman famously wrote that the mystery of wave-particle duality in the double-slit

[Infinitesimals]

What Are Infinitesimals – Advanced Version

May 24, 2023

/

Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say

Popular Science Communication: Bridging Science & Public

April 30, 2023

/

Abstract Famous Science Communicators There is so much to say about the many endeavors by professional scientists to explain to us the world. The list

art of integration

[art of integration]

The Art of Integration

April 10, 2023

/

Abstract My school teacher used to say "Everybody can differentiate, but it takes an artist to integrate." The mathematical reason behind this phrase

teaching physics

[teaching physics]

A Lesson In Teaching Physics: You Can’t Give It Away

March 16, 2023

/

A central principle of Physics Forums regarding homework help is not to provide solutions on demand but to guide students along a path to the answer. The

integration and complex differentiation

[integration and complex differentiation]

An Overview of Complex Differentiation and Integration

March 14, 2023

/

Abstract I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments

Measure Internal Resistance of Battery

[Measure Internal Resistance of Battery]

How to Measure Internal Resistance of a Battery

February 14, 2023

/

Introduction A commonly encountered school-level Physics practical is the determination of the internal resistance of a battery - typically an AA or D

lie group physics

[lie group physics]

When Lie Groups Became Physics

February 8, 2023

/

Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham

Why There Are Maximum Mass Limits for Compact Objects

January 27, 2023

/

In this article, we will look at why there are maximum mass limits for objects that are supported against gravity by degeneracy pressure instead of kinetic

gravity collapse

[gravity collapse]

Oppenheimer-Snyder Model of Gravitational Collapse: Implications

January 15, 2023

/

Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn the last article in this series, we finished up with a metric for the Oppenheimer-Snyder

tensors relativity

[tensors relativity]

Tensors Explained: Beginner’s Essentials for Relativity

January 11, 2023

/

If you try learning general relativity, and sometimes special relativity, on your own, you will undoubtedly run into tensors. This article will outline

gravity collapse

[gravity collapse]

Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details

January 9, 2023

/

Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn a previous article, I described in general terms the model of gravitational

Twin Paradox Guide: Spacetime Geometry Explained Clearly

January 2, 2023

/

Introduction This article is intended for anyone who wants to start a thread here at Physics Forums on the twin paradox. There are already many, many threads

gravity collapse

[gravity collapse]

The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview

December 28, 2022

/

Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsMost people who have spent any time at all studying GR are familiar with the

object slide down ramp physics

[object slide down ramp physics]

Subtleties Overlooked in Friction Questions: Object Slides Down Ramp

December 6, 2022

/

Problem statement (simplified) An object slides down a ramp at angle to encounter level ground. Both surfaces have kinetic friction: ' on the ramp,

math classifications

[math classifications]

Mathematics Fields & Applications: A Structured Guide

October 25, 2022

/

Introduction I often read questions about our classification scheme that we use on physicsforums.com to sort posts by science fields and subjects, what

Reduction of Order For Recursions

October 4, 2022

/

This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.Most of us remember recursion

history of numbers

[history of numbers]

Counting to p-adic Calculus: All Number Systems That We Have

September 18, 2022

/

An entire book could easily be written about the history of numbers from ancient Babylon and India, over Abu Dscha'far Muhammad ibn Musa al-Chwarizmi (##\sim

evariste galois

[evariste galois]

variste Galois and His Theory

September 5, 2022

/

* Oct. 25th, 1811 May 31st, 1832 ... or why squaring the circle is doomed. Galois died in a duel at the age of twenty. Yet, he gave

definition differences

[definition differences]

Yardsticks to Metric Tensor Fields

August 27, 2022

/

I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist,

Program ATmega8A with Arduino: Step-by-Step Guide

August 13, 2022

/

Introduction If you are interested in programming and electronics, you probably do not need an introduction to Arduino. If you want to make your Arduino

P vs. NP and what is a Turing Machine (TM)?

July 25, 2022

/

P or NP This article deals with the complexity of calculations and in particular the meaning of ##P\stackrel{?}{\neq}NP## Before we explain what P and

quantum computers 101

[quantum computers 101]

Quantum Computing Explained: Qubits, Algorithms & Applications

July 14, 2022

/

Overview This article provides an accessible introduction to quantum computing, a cutting-edge technology that processes certain types of information

gauss law misconceptions

[gauss law misconceptions]

A Physics Misconception with Gauss’ Law

June 12, 2022

/

Introduction It is relatively common to see the following type of argument: The surface area is ##A## and the enclosed charge is ##Q##. The electric

How to Model a Magnet Falling Through a Conducting Pipe

June 11, 2022

/

Introduction In an earlier article, we examined a magnet falling through a solenoid. We argued that the point dipole model can account for the basic features

How to Model a Magnet Falling Through a Solenoid

June 5, 2022

/

Introduction Modeling a magnet realistically is a task best done numerically. Even the simplified model of two separated disks with uniform surface

why can we jump

[why can we jump]

How Can We Jump When the Ground Does No Work?

May 30, 2022

/

Common mistaken argument It is relatively common on Physics Forums to see arguments that are effectively similar to the following: When we jump off the

Riemann Hypothesis History

[Riemann Hypothesis History]

The History and Importance of the Riemann Hypothesis

May 21, 2022

/

Riemann Hypothesis History The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field

magnetism current

[magnetism current]

Symmetry Arguments and the Infinite Wire with a Current

May 17, 2022

/

Many people reading this will be familiar with symmetry arguments related to the use of Gauss law. Finding the electric field around a spherically symmetric

cpu programming

[cpu programming]

Parallel Programming on a CPU with AVX-512

April 23, 2022

/

This article is the second of a two-part series that presents two distinctly different approaches to parallel programming. In the two articles, I use different

programming gpu

[programming gpu]

CUDA Regression Line: GPU Parallel Programming Guide

April 16, 2022

/

Overview This article is the first of a two-part series that presents two distinctly different approaches to parallel programming. In the two articles,

hypercube integral

[hypercube integral]

A Novel Technique of Calculating Unit Hypercube Integrals

April 14, 2022

/

Introduction In this insight article, we will build all the machinery necessary to evaluate unit hypercube integrals by a novel technique. We will first

Riemann Hypothesis

[Riemann Hypothesis]

The Extended Riemann Hypothesis and Ramanujan’s Sum

April 4, 2022

/

Riemann Hypothesis and Ramanujan's Sum ExplanationRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All

The Amazing Relationship Between Integration And Euler’s Number

March 28, 2022

/

We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that

mermin device quantum entanglement

[mermin device quantum entanglement]

Superdeterminism and the Mermin Device

March 20, 2022

/

Superdeterminism as a way to resolve the mystery of quantum entanglement is generally not taken seriously in the foundations community, as explained in

relativity rotated graph

[relativity rotated graph]

Relativity on Rotated Graph Paper (a graphical motivation)

February 16, 2022

/

(based on https://www.physicsforums.com/threads/teaching-sr-without-simultaneity.1011051/post-6588952 and https://physics.stackexchange.com/a/689291/148184

probabilities virus testing

[probabilities virus testing]

Probabilistic Factors Involved in Disease and Virus Testing

February 6, 2022

/

Introduction This Insight looks at the various probabilistic factors and related terminology involved in disease and virus testing.As we all know,

quantum mechanics new approach

[quantum mechanics new approach]

Quantum Physics via Quantum Tomography: A New Approach to Quantum Mechanics

January 8, 2022

/

This Insight article presents the main features of a conceptual foundation of quantum physics with the same characteristic features as classical physics

trig special functions

[trig special functions]

A Trick to Memorizing Trig Special Angle Values Table

January 1, 2022

/

In calculus classes when you are asked to evaluate a trig function at a specific angle, it's 99.9% of the time at one of the so-called special angles we

setup raspberry pi cluster

[setup raspberry pi cluster]

Raspberry Pi Cluster Guide: Parts, MPI & Slurm Tips

December 22, 2021

/

INTRODUCTION As a long-time computer programmer and almost as long a High-Performance Computer (HPC) user, I didn't know anything about how these machines

Theorema Primum tutorial

[Theorema Primum tutorial]

An Introduction to Theorema Primum

November 27, 2021

/

Introduction Whilst no doubt most frequenters of "Physics Forums" will be familiar with Nicolaus Copernicus as the scientist who advanced the (at the

python sympy module

[python sympy module]

Python’s Sympy Module and the Cayley-Hamilton Theorem

November 9, 2021

/

Two of my favorite areas of study are linear algebra and computer programming. In this article I combine these areas by using Python to confirm that a

[Kerr Spacetime]

Geodesic Congruences in FRW, Schwarzschild and Kerr Spacetimes

October 6, 2021

/

Introduction The theory of geodesic congruences is extensively covered in many textbooks (see References); what follows in the introduction is a brief

Physical Applications of the “Tan Rule”

September 20, 2021

/

Introduction Every secondary school student who has encountered trigonometry in his/her Math syllabus will most likely have come across the sine, cosine,

MIT course corrections

[MIT course corrections]

Corrections to MIT Open Courseware: Systems of Varying Mass

September 14, 2021

/

Corrections to MIT Open Courseware 8-01sc classical mechanics, fall 2016 Applying Newton's Laws to systems of Varying Mass PDF Course Link My concerns

quantum information

[quantum information]

How Quantum Information Theorists Revealed the Relativity Principle at the Foundation of Quantum Mechanics

July 25, 2021

/

This Insight is a condensed version of No Preferred Reference Frame at the Foundation of Quantum Mechanics. Reference numbers here correspond to that paper.

Investigating Some Euler Sums

July 21, 2021

/

So, why only odd powers? Mostly because the even powers were solved by Leonard Euler in the 18th century. Since the mathematical toolbox at that

Quaternions in Projectile Motion

[Quaternions in Projectile Motion]

Quaternions in Projectile Motion

July 15, 2021

/

Introduction In a previous Physics Forums article entitled How to Master Projectile Motion Without Quadratics, PF user@kurumanbrought to our

quantum double slit experiment

[quantum double slit experiment]

Quantum Mechanics and the Famous Double-slit Experiment

July 1, 2021

/

What Does the Double-Slit Experiment Prove About Quantum Reality?The double-slit experiment demonstrates that light and matter exhibit both wave and

Safety Factors: Why Forums Avoid Design Advice Online

June 26, 2021

/

Forum Guidelines There are often questions on Physics Forums related to building machines or structures that are left unanswered because of this guideline

LCEVC and EVC: Efficient Video Encoding Guide for 8K

June 19, 2021

/

Introduction Background: H.264 and H.265 First, for background, here is an overview of the most common video compression standard in use today, H264:

solving projectile motion

[solving projectile motion]

How to Solve Projectile Motion Problems in One or Two Lines

June 8, 2021

/

Introduction We show how one can solve most if not all, introductory-level projectile motion problems in one or maybe two lines. To this end, we forgo

10 Math Things We All Learnt Wrong At School

April 21, 2021

/

Some of these could even lead to heavy debates within the scientific community, so maybe I should say: from my point of view. So before you get excited

Fourier Series Riemann Zeta Function

[Fourier Series Riemann Zeta Function]

Computing the Riemann Zeta Function Using Fourier Series

April 9, 2021

/

Euler's amazing identity The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous

tolman law beginner

[tolman law beginner]

Tolman Law in a Nutshell

April 2, 2021

/

The Tolman law describes how the temperature in a fixed gravitational field depends on the position (see https://arxiv.org/abs/1803.04106 for a pedagogic

physics cannonball projectile

[physics cannonball projectile]

Maximizing Horizontal Range of a Projectile

March 29, 2021

/

Introduction A recent homework problem that appeared in the forums was concerned with maximizing the horizontal range of a projectile subject to the launch

electromagnetic computations duality

[electromagnetic computations duality]

Building a Numerical FEM Solver for Duality in EM Fields

March 18, 2021

/

In the previous insights article (How to Use Duality in Computational Electromagnetic Problems), I covered some uniqueness theorems for the Riemann-Silberstein

Electric Field Seen by an Observer

[Electric Field Seen by an Observer]

The Electric Field Seen by an Observer: A Relativistic Calculation with Tensors

March 13, 2021

/

This Insight was inspired by the discussion in "electric field seen by an observer in motion", which tries to understand the relation between two expressions:

valentines reflections graphs

[valentines reflections graphs]

Valentine’s Reflections: Mathematical Matters of the Heart

March 1, 2021

/

Introduction Being a somewhat geeky Maths 'nerd', I spent days leading up to Valentine's Day trying to find a Maths function appropriate to the day. In

Bayesian statistics part 4

[Bayesian statistics part 4]

Posterior Predictive Distributions in Bayesian Statistics

February 17, 2021

/

Confessions of a moderate Bayesian, part 4 Bayesian statistics by and for non-statisticiansRead part 1: How to Get Started with Bayesian StatisticsRead

electromagnetic computations

[electromagnetic computations]

How to Use Duality in Computational Electromagnetic Problems

February 14, 2021

/

Some weeks ago I happened across a post that caught my eye. Dale asked a question about the number of photons in an electromagnetic field. His question

impossible triangles

[impossible triangles]

Ambiguous SSA Case in Triangles Law of Sines Explained

January 28, 2021

/

What is the ambiguous case? In U.S. high schools, students are often introduced to geometric proofs for the first time. A common task is to use a small

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