Physics, Math and Science, Explained
Revisiting the Velocity-Time Function
/1 Comment/in Mechanics, Physics Tutorials/by Neil ParkerKinematics: 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…
Trending Articles
What Planck Length Is and It’s Common Misconceptions 178 views | by Alex Klotz | posted on September 9, 2015
Can We See an Atom? 128 views | by Alex Klotz | posted on September 28, 2015
The Block Universe – Refuting a Common Argument 78 views | by PeterDonis | posted on November 13, 2015
Self-Study High School Math: Best Books & Order to Learn 74 views | by Micromass | posted on November 19, 2015
Why You Can’t Quantum Tunnel Through a Wall 69 views | by ZapperZ | posted on August 10, 2015
Recent Entries

Do Black Holes Really Exist?
/
68 Comments
Introduction
The purpose of this article is to discuss the title question from several different viewpoints, to show that it isn't as simple as it looks.…

Frequently Made Errors in Climate Science – The Greenhouse Effect
1.What is meant by "The Greenhouse Effect"?
Many gases, such as H2O, CO2, CH4, are transparent to visible light but absorb and emit parts of the infrared…

Learn About the Speed of Light and Galilean Relativity
Introduction
In this article, I will discuss some experiments in the 19th and early 20th centuries that looked at how the velocity of a light source affects…

Ice Quiz: All About Ice Trivia
Ice is the solid form of water, and understanding its basic properties (freezing point, density, crystal structure) helps explain everything from why it…

What Causes Cancer: Bad Luck or Bad Lifestyles?
What causes cancer?
For the most part, cancer is a disease that arises from mutations in the body that accumulate over time. These mutations knock…

Understanding Buoyancy: The Limits of Archimedes
It’s a principle as old as science itself, and remarkably, remains essentially unchanged in the light of revolution after revolution in our understanding…

Why the Observable Universe Is Bigger Than Its Age
The observable universe has a radius of about 46 billion light-years, far larger than its age of roughly 14 billion years. This happens because space itself…

What Is Heat? The Real Thermodynamics Definition Explained
Heat is not something a body possesses. Heat is a transfer mechanism: the non-mechanical exchange of energy between a system and its surroundings that…

How to Solve Nonhomogeneous Linear ODEs using Annihilators
My previous Insights article, Solving Homogeneous Linear ODEs using Annihilators, discussed several examples of homogeneous differential equations, equations…

What Is the Steady-state Model and Why It’s No Longer Viable
Introduction to the Steady State Model
Back in 2011, I wrote a FAQ entry for Physics Forums on the steady state model and why it is no longer viable.…

Solving Homogeneous Linear ODEs using Annihilators
In this Insights article we'll look at a limited class of ordinary differential equations -- homogeneous linear ODES with constant coefficients. Although…

Trials and Tribulations of a Physicist who Became a Math Geek
How did I go from the brink of changing my major from physics to ceramics (no more math) to the Math faculty of the Air Force Academy? How did I go from…

Learn About Matrix Representations of Linear Transformations
Let X and Y be finite-dimensional vector spaces. Let ##T:X\to Y## be a linear transformation. Let ##A=(e_1,\dots,e_n)## and ##B=(f_1,\dots,f_m)## be ordered…

Why Don’t Electrons Crash Into the Nucleus? Explained
Atoms are stable because electrons do not behave as classical orbiting particles. Quantum mechanics describes electrons using wavefunctions and stationary…

How Does an Airplane Wing Work? A Primer on Lift
Many people ask how an airplane wing works, and several answers are commonly given. You might have even seen vigorous arguments between proponents…

Why the Big Bang Wasn’t a Black Hole: Cosmology Explained
The Big Bang was not a black hole because it happened everywhere in space at once rather than at a single point, and because the near-perfect homogeneity…

Simple Python Debugging with Pdb: Part 2
This Insight article is the continuation of the first article, Simple Python Debugging with Pdb: Part 1.In this article, let's look at another important…

Learn Simple Python Debugging with Pdb
I'm pretty new to Python, so I was looking around for some debugging tools. At first, I dismissed Pdb (Python debugger) as being too primitive, but after…

A Guide to Going Back to School For Math or Science
Motivations for creating the series
I came to Physics Forums originally to read ZapperZ’s thread about becoming a physicist back in 2013. It was one…

Learn About Energy Gained by Charge in an Electrostatic Field
Energy gain of an electron in a potential V
I have seen this question being asked frequently on here. A lot of students have a bit of an issue in understanding…

Self-Study High School Math: Best Books & Order to Learn
To self-study high school mathematics in a logical sequence, most learners should progress through basic algebra, then synthetic geometry, then trigonometry,…

How to Self-Study Calculus: Topics and Book Guide
To self-study calculus, work through four core areas in order: differentiation, integration, sequences and series, then multivariable calculus. Each builds…

How to Self-Study Mathematics: A Step-by-Step Guide
Self-studying mathematics is achievable for most learners willing to commit to daily, consistent practice rather than occasional cramming. Success depends…

Have Scientists Seen an Electron? How We Detect Them
Have Scientists Seen an Electron? Why Seeing Isn't Enough
This article is not about literally seeing a single electron with your eye; it addresses the…

The Block Universe – Refuting a Common Argument
The "block universe" interpretation of SR has come up repeatedly in threads here on PF. Rather than link to them, I want to summarize a common argument…

A Geometrical View of Time Dilation and the Twin Paradox
Based on the number of questions we receive on the topic at Physics Forums, there is a lot of confusion in the general public about how time dilation works…

Understanding Retrocausality and Blockworld
In an Insights series “Blockworld and its Foundational Implications,” I quoted Huw Price and Ken Wharton in several places advocating a blockworld…

Is the Universe Rotating? What Physics Tests Reveal
All current astronomical observations are consistent with zero global rotation of the universe. Solar-system data set a model-independent upper limit of…

Blockworld and its Foundational Implications: Delayed Choice and No Counterfactual Definiteness
In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity. In part 3, I introduced…

Blockworld and its Foundational Implications: General Relativity and Closed Timelike Curves
In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity (SR). In part 3, I introduced…

Blockworld and its Foundational Implications: General Relativity and the Big Bang
In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity (SR). Geroch sums up the…

Blockworld and Its Foundational Implications: The Relativity of Simultaneity and Blockworld
In part 1 of this 5-part Insights series, I introduced two consequences of the second postulate of special relativity (SR): time dilation…

Blockworld and Its Foundational Implications: Time Dilation and Length Contraction
This is the first in a 5-part series of Insights that will introduce blockworld (aka "block universe”) and use it to address a puzzle…

Atomic Positioning with DNA Hinges
I was perusing the Nature Nanotechnology website today when I came across an interesting article by Funke and Dietz, called "Placing molecules with Bohr…

Examples of Prequantum Field Theories IV: Wess-Zumino-Witten-type Theories
At the end of this page, we will have come full circle back to the first article in the series 20 years ago -- The M-Theory Conjecture.We…

Examples of Prequantum Field Theories III: Chern-Simons-type Theories
After having constructed gauge fields and higher gauge fields in the previous article by systems of ##L_\infty##-algebroid-valued…

Does Heisenberg’s Uncertainty Principle Break Energy Conservation?
Direct answer: No, the Heisenberg time-energy uncertainty relation does not violate conservation of energy in quantum mechanics. The relation describes…

Examples of Prequantum Field Theories II: Higher Gauge Fields
After having recalled ordinary gauge fields from a dg-algebraic perspective in the previous article, here I discuss…

Examples of Prequantum Field Theories I: Gauge Fields
After having motivated the need for prequantum field theory and having laid out its principles (i. extremal action, ii. global…

Fun with Self-Avoiding Walks Simulations
This post is about some simulations I did of self-avoiding random walks. These are what they sound like with each step, the position of the walk moves…

Why 1 = 0.999…: Four Simple Proofs Explained
Yes, 1 and 0.999... are exactly the same real number, not merely two numbers that happen to be extremely close. This surprises many people because 0.999...…

Higher Prequantum Geometry V: The Local Observables – Lie Theoretically
This article discusses how the previous considerations naturally follow the concepts of local observables of local field theories and of…

Do Photons Move Slower in a Solid Medium?
Scope and context
This question appears often because it has been shown that in a normal, dispersive solid such as glass, the speed of light is slower…

Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively
The Euler-Lagrange ##p##-gerbes discussed in the previous article are singled out as being exactly the right coherent refinement…

Higher Prequantum Geometry III: The Global Action Functional – Cohomologically
The previous article ended with the concept of classical locally variational field theories, of which a class of examples are…

A Poor Man’s CMB Primer: Bumps on a Blackbody
Astronomers Arno Penzias and Robert Wilson discovered the cosmic microwave background in 1965. They were not looking for it.They were…

Higher Prequantum Geometry II: The Principle of Extremal Action – Comonadically
The previous article motivated the importance of considering "pre-quantum field theory" in-between classical and quantum field…

Time Dilation & Redshift of Schwarzschild Black Holes
Time dilation near a Schwarzschild black hole
The following is an overview of the time-dilation and gravitational-redshift effects of a static (Schwarzschild)…

Higher Prequantum Geometry I: The Need for Prequantum Geometry
Before proceeding with a discussion of the super p-brane sigma models, whose emergence from the superpoint I discussed in the previous article,…

Can We See an Atom?
Introduction
In this article I discuss how images of atoms are made, what exactly we are looking at, and what it means to "see" an atom. Over the decades…

Does Gravity Gravitate: The Wave
In the first two posts in this series, we looked at different ways of interpreting the question "does gravity gravitate?" We left off at…

Why a Rigid Rod Can’t Send Faster-Than-Light Signals
A perfectly rigid rod cannot send faster-than-light (FTL) signals because no such rod exists. Every real material transmits mechanical disturbances at…

Does Gravity Gravitate? Part 2
In the first post of this series, I talked about two ways to answer the title question, one leading to the answer "no" and the other leading…

Proofs in Mathematics: Methods, Examples, and How to Write Them
A mathematical proof is a logical argument, built only from definitions, axioms, and previously established results, that shows a statement is true in…

PF’s policy on Lorentz Ether Theory and Block Universe
What is the PF's policy on Lorentz Ether Theory and Block Universe?Debates about the superiority or "truth" of modern Lorentz Ether Theory (LET) and…

Doubt vs. Discouragement: A Physics Student’s Calculus Story
Overcoming discouragement means recognizing that another person's doubt about your ability is not evidence of your limits. The author failed a calculus…

Does Gravity Gravitate?
The title question of this article is one that often comes up in PF threads, and I would like to give my take on it. This will be the first…

Pluto Quiz: Test Your New Horizons Mission Knowledge
This quiz tests reader knowledge of NASA's New Horizons mission to Pluto, covering key facts and discoveries reported during the mission's flyby coverage.…

A Poor Man’s CMB Primer: The Birth of a Cosmic Background Radiation
The early universe was hot. So hot that nuclei boiled. The great thermal energy of the universe overwhelmed the confining efforts of the…

Mixing vs Beating: Superheterodyne & Beat Frequency
Mixing vs. Beating: a brief history
A long time ago I read a paper in the IEEE Proceedings recounting the history of the superheterodyne receiver. Overall…

The Balloon Analogy Explained: Cosmic Expansion Without a Center
The balloon analogy uses pennies glued to an inflating balloon to show two ideas from modern cosmology: that unbound regions of the universe expand roughly…

Why Is Acceleration Due to Gravity a Constant?
This question has popped up many times. So here is an attempt to address it.To answer this question at the elementary level, several assumptions will…

Struggles With the Continuum: Point Particles and the Electromagnetic Field
In these posts, we're seeing how our favorite theories of physics deal with the idea that space and time are a continuum, with points described…

Explosion-Generated Collapsing Vacuum Bubbles Reach 20,000 Kelvin
Experiment overview
In a recent paper published in Physical Review Letters (PRL), one of the top physics journals, Jérôme Duplat and Emmanuel Villermaux…

How Evolution Works: Variation, Inheritance, Selection
Evolution is the process by which populations of organisms change over generations through three interacting mechanisms: random genetic variation, inheritance…

Cardinal and Ordinal Numbers: An Informal Introduction
Cardinal numbers measure the size of a set, including infinite sets, while ordinal numbers describe the structure of a well-ordered set. Every set can…

A Poor Man’s CMB Primer: Orientation of the Universe
This is a picture of the cosmic microwave background:Fig 1. The cosmic microwave background as seen by the European Space Agency's…

Why Does C Have a Particular Value, and Can It Change?
Short answer:
Because c (speed of light) has units, its value is what it is only because of our choice of units, and there is no meaningful way to test…

What Planck Length Is and It’s Common Misconceptions
The Planck length is an extremely small distance constructed from physical constants. Many misconceptions overstate its physical significance, claiming…

Struggles With The Continuum: Quantum Mechanics of Charged Particles
Last time we saw that nobody yet knows if Newtonian gravity, applied to point particles, truly succeeds in predicting the future. To be precise:…

How to Overcome Procrastination: 8 Proven Methods
Procrastination is a predictable pattern of avoidance behavior, not simply laziness or poor time management. It splits into two main types: anxious procrastination,…

How Fast Do Changes in the Gravitational Field Propagate?
Overview
General relativity predicts that disturbances in the gravitational field propagate as gravitational waves, and that low-amplitude gravitational…

Why Imagination Without Knowledge Is Ignorance Waiting to Happen
Introduction
My feelings on people who think that imagination is more important than knowledge are well-known. These people simply are parroting Einstein's…

Animal Speed Scaling: Body-Lengths per Second Across Sizes
Introduction
In a recent American Journal of Physics issue, I read an interesting paper by Nicole Meyer-Vernet and Jean-Pierre Rospars examining the top…

Infinity in Mathematics: What It Really Means, Explained
Infinity is not a real number but a set of distinct mathematical tools, including the extended real line, the projective real line, the Riemann sphere,…

Apollo Moon Missions: Eugene Cernan and the Final Landing
Eugene Cernan became the last person to walk on the moon on December 11, 1972, during NASA's Apollo 17 mission, closing out a nine-mission lunar exploration…

Is Spacetime Really a Continuum?
Introduction
Is spacetime really a continuum? That is, can points of spacetime really be described---at least locally---by lists of four…

“Violating” Einstein’s Photoelectric Effect Model
Einstein's Photoelectric Insight
One of the most spectacular theoretical descriptions that Einstein had ever produced is the corpuscular nature of light…

Is 1 Equal to 0.999…? Rigorous Proof Explained
Yes, 1 equals 0.999... exactly, not approximately. This is a proven mathematical fact once the notation 0.999... and the concept of an infinite sum are…

Inflationary Misconceptions and the Basics of Cosmological Horizons
Introduction
It is a common saying that during inflation "space expanded faster than the speed of light." This statement is meant to articulate the…

The Concept of Zero: History, Rules, and Why 0/0 Fails
Zero is neither positive nor negative, is defined as the additive identity so that x + 0 = x for every number x, and produces 0 when multiplied by any…

What Happens When You Flip the Light Switch: From Zero Seconds to 20 Years
Introduction
Few people know what happens when a customer does something like flipping a light switch. The full explanation requires descriptions of topics…

Will All Matter Be Converted to Photons?
Will all matter be converted into black holes and then photons?
No. There is a misconception that a lot of laypeople seem to be picking up recently from…

Do Photons have Mass?
Do photons have mass?
The quick answer: NOCommon source of confusion
However, this is where it gets a bit confusing for most people. This is because,…

Learn the Relativistic Work-Kinetic Energy Theorem
I was bothered for a long time by the reasons for the relativistic validity of the work-kinetic energy relation ##\Delta E=Fd##, which holds without any…

Emergence From the Superpoint
In the previous article we saw that the generalization of super-Lie algebras to homotopy super-Lie n-algebras (super L-infinity algebras)…

An Equation for the Centrifugal Force Reversal Near A Black Hole
My goal in this article is to derive a simple equation for the proper acceleration of an observer traveling on a circular path around a Schwarzschild black…

Homotopy Lie-n Algebras in Supergravity
The previous article in this series claimed that the mathematics of the 21st century that had fallen into the 1970s in the form of string…

Learn A Short Proof of Birkhoff’s Theorem
Birkhoff's theorem is a very useful result in General Relativity, and pretty much any textbook has a proof of it. The one I first read was in Misner, Thorne,…

It Was 20 Years Ago Today — the M-theory Conjecture
While the world didn't end, after all, 15 years back at the turn of the millennium, in hindsight it is curious that, almost unnoticed,…

Why You Should Not Use Wikipedia As Your Primary Source
It is no secret to anyone who has read my posts in this forum for a while that I do not like Wikipedia. I think that there is a fundamental flaw with the…

Bell Spaceship Paradox: Why the String Breaks in SR
Bell spaceship paradox — setup
Bell describes two spaceships that start out at rest relative to each other with an elastic string between them, one…

Learn a Misconception of the Heisenberg Uncertainty Principle
One of the common misconceptions about the Heisenberg Uncertainty Principle (HUP) is that it is the fault of our measurement accuracy.A description…

Why You Can’t Quantum Tunnel Through a Wall
Can You Quantum Tunnel Through a Wall?
Short answer
The short and sweet answer if a tennis ball, a bowling ball, or any other kind of ordinary macroscopic…

What is relativistic mass and why it is not used much?
It happens that the term relativistic mass is used, in particular in the introductory text on special relativity. It should be noted that whether or not…

The Most Important Thing You Can Learn from Physics Forums
Introduction
It is a bit pompous of me to think that I can tell you the most important thing you can learn from Physics Forums (PF). After all, each…

Plus/minus What? How to Interpret Error Bars
People sometimes find themselves staring at a number with a ± in it when a new physics result is presented. But what does it mean? This Insight aims to…

Approximate LCDM Expansion in Simplified Math (Part 4)
Part 4: Cosmic Recession Rates
An astronomer, accompanied by his amateur relativist friend, aimed a telescope at a distant galaxy and measured…

False Physics Major Dichotomy: Theorist or Experimentalist?
Figure showing links between ballistic pressure waves and traumatic brain injury.Many physics majors believe that pursuing a Ph.D. in…

CV Tips for Physicists: Secure Postdoc & Research Jobs
Full Chapter List - So You Want To Be A Physicist... Series
Part I: Early Physics Education in High schools
Part II: Surviving the First Year…

