Physics, Math and Science Articles

Velocity Time Function

Revisiting the Velocity-Time Function

/0 Comments/in /by

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…

Trending Articles

Recent Entries

entropy

How to Determine the Change in Entropy

/
9 Comments
How do you determine the change in entropy for a closed system that is subjected to an irreversible process?Here are some typical questions we get…
Supersymmetry

Learn About Supersymmetry and Deligne’s Theorem

/
54 Comments
 In 2002, Pierre Deligne proved a remarkable theorem on what mathematically is called Tannakian reconstruction of tensor categories. Here…
Schwarzschild

Learn Orbital Precession in the Schwarzschild and Kerr Metrics

/
7 Comments
The Schwarzschild Metric A Lagrangian that can be used to describe geodesics is [itex]F = g_{\mu\nu}v^\mu v^\mu[/itex], where [itex]v^\mu = dx^\mu/ds[/itex]…
spacetimetetrad

Tetrad Formalism: Ricci Coefficients & Riemann Tensor

/
1 Comment
Tetrads and Notation A spacetime is often described in terms of a tetrad field, that is, by giving a set of basis vectors at each point. Let the vectors…
LHC4

LHC Part 4: Searching for New Particles and Decays

/
17 Comments
 The LHC experiments are in full swing collecting data this year (more information in the forum), and a while ago the collaboration of the…
acusticbeats

Mathematics of Acoustic Beats: Explanation & Examples

/
11 Comments
Introduction Purpose and scope In late high school physics courses and first-year university courses, the phenomenon of acoustical 'beats' (digital audio)…
omissionguage

Why the Gauge (Henstock-Kurzweil) Integral Matters

/
31 Comments
The gauge integral (Henstock–Kurzweil) The current (pure) mathematics curriculum at the university is well-established. Most of the choices made are…
polymer

DNA Mapping & Nanopore Sequencing: Nanochannel Insights

/
4 Comments
Introduction A single DNA molecule snaking its way through an array of cavities. This movie is from one of my later Ph.D. experiments. …

Learn About Relativity on Rotated Graph Paper

/
15 Comments
This Insight is a follow-up to my earlier tutorial Insight (Spacetime Diagrams of Light Clocks). I gave it a different name because I am placing more…
Brachistochrone

Explaining the General Brachistochrone Problem

/
3 Comments
Consider a problem about the curve of fastest descent in the following generalized statement. Suppose that we have a Lagrangian system $$L(x,\dot x)=\frac{1}{2}g_{ij}(x)\dot…
qualitycontrol

Hardware, Software, and C Programming Quality Tips

/
6 Comments
On Hardware Quality Designing Quality vs Testing It is impossible to test quality into a product. Quality must be designed into the product. Production…
integralmisconceptions

Integration Paradoxes: Indefinite Integrals and Constants

/
40 Comments
Introduction Integration is an incredibly useful technique taught in all calculus classes. Nevertheless, there are certain paradoxes involved with integration…
basickinematics

Learn Basic Kinematics in Classical Mechanics

/
8 Comments
There is an interesting thing in teaching of Classical Mechanics. Several theorems which presented below form a core part of kinematics for all Russian…
groupsandgeometry

Klein’s Erlangen Program: Groups Define Geometry

/
7 Comments
Klein's Erlangen program: groups and geometry There is a very deep link between group theory and geometry. Sadly, this link is not emphasized in most…
selfstudy

Self-Study Roadmap: Abstract Algebra, Groups to Galois

/
4 Comments
Overview Roadmap There are three major areas of mathematics: geometry, analysis, and algebra. This insight gives a roadmap for learning basic abstract…
computers

Ramsey Theory: Foundations, Generalizations, Key Results

/
5 Comments
Ramsey theory has its origins in a very nice riddle Consider a party of 6 people. Any two of these six will either be meeting each other for the first…

Learn About Spacetime Diagrams of Light Clocks

/
25 Comments
We demonstrate a method for constructing spacetime diagrams for special relativity on graph paper that has been rotated by 45 degrees. Many quantitative…
kinematics

Presenting a Rare Kinematic Formula

/
1 Comment
Here we present some useful kinematic fact which is uncommon for textbooks in mechanics. Consider a convex rigid body (RB) rolling without slipping…
universestart

Did the Big Bang Have a Center? Explaining Cosmology

/
28 Comments
Did the Big Bang Have a Center? What the standard models say According to standard cosmological models based on general relativity — and which agree…
nugatory

Interview with a Physics Mentor: Nugatory

/
5 Comments
Give us a little background on yourself? I was born in Washington DC, grew up in Texas, bachelor's in physics from Harvard in 1978 but found a career…

A Black Hole Problem From Greg Egan’s “Incandescence”

/
8 Comments
The following problem was inspired by reading Greg Egan's "Incadescence".  Consider a body, orbiting a Schwarzschild black hole and tide locked to it,…
angularvelocity

Elementary Construction of the Angular Velocity

/
1 Comment
Physics books seldom contain an accurate definitions of the angular velocity of a rigid body. I believe that the following construction is as simple as…
softwarenotperfect

Why Your Software is Never Perfect

/
52 Comments
Introduction We occasionally have students ask for help on software, "My software is perfect, but it doesn't work!" Your software is never perfect. My…
selfstudy

Why Self-Studying Math Often Fails: 5 Common Pitfalls

/
49 Comments
My experience helping self-studiers For several years I have helped students who were self-studying mathematics. I did this free, without compensation,…
irrationalnumbers

Mathematical Irrationality for Dummies

/
13 Comments
On one of my restless wanderings around the Internet, I came upon a collection of proofs of the statement that the square root of 2 is an irrational…
Photoacoustic

Light and Sound Interactions: Photoacoustic & Acousto-Optic

/
6 Comments
Introduction I recently wrote a post on my blog about a fairly esoteric idea regarding sound propagating through light. This inspired me to write an…
logicp3

Scientific Inference: Balancing Predictive Success with Falsifiability

/
9 Comments
  Bayes' Theorem: Balancing predictive success with falsifiability Despite its murky logical pedigree, confirmation is a key part of learning.…
seancarroll

Sean Carroll Interview: Cosmology, Books & Insights

/
10 Comments
We are pleased to introduce Sean Carroll. Sean Carroll is a theoretical physicist at Caltech specializing in cosmology and quantum mechanics. He has a…
physicsathome

High School Physics: Courses, Resources, and Prep Guide

/
2 Comments
Why High School Physics Matters A year-long high school Physics course is important preparation for all college majors in science, engineering, and medicine…
themoon

Moon Quiz: Test Yourself on the Moon

/
36 Comments
For ages it has been Earth's little sister. Find out just how much you know about the Moon and our relationship with it. Please post your score in the…
selfstudyanalysis

Advanced Real Analysis: Measure & Functional Study Plan

/
14 Comments
Prerequisites If you wish to follow this guide, you should be familiar with analysis on ##\mathbb{R}## and ##\mathbb{R}^n##. See my previous insight for…
mathjax

Peter Krautzberger on MathJax — Accessibility & Future

/
3 Comments
We are pleased to introduce Peter Krautzberger. Peter is a manager for MathJax, the technology behind the display of equations on Physics Forums. "The…
logicp2

Scientific Inference: Do We Really Need Induction?

/
14 Comments
  Part 2. "We don't need no stinkin' induction" -- Sir Karl Popper The traditional scientific method supposedly employs induction both in the…
linearalgebra_selfstudy

Linear Algebra Roadmap: Prereqs, Books & Topics Guide

/
11 Comments
Overview This article gives a roadmap for students to learn the basics of linear algebra. Aside from calculus, linear algebra is one of the most applicable…
antigravity

Roger Babson & the Gravity Research Foundation History

/
13 Comments
Roger Babson and His War on Gravity In the early 20th century, an eccentric businessman named Roger Babson declared gravity to be mankind's greatest…
mathresources

Top High School College-Prep Math Options & Resources

/
13 Comments
The popularity of homeschooling and failures of the public schools often present the need for parents to teach or supplement college preparatory math courses…
logic1

Hume and the Problem of Induction in Science Explained

/
39 Comments
Click for Full SeriesPart 1: Logical induction ain’t logical Part 2: We don’t need no stinkin’ induction Part 3: Balancing Predictive Success…
drclaude

Interview with Chemical Physicist: DrClaude

/
5 Comments
DrClaude is a mentor at Physics Forums!What brought you to Sweden from Canada? Life, the Universe, and Everything. While in Canada, I was offered the…
peterdonis

PeterDonis Interview – Physics Mentor on Science & Tech

/
8 Comments
PeterDonis is a Physics Mentor at Physics Forums. About PeterDonis I have been interested in science since childhood. I hold two degrees in Nuclear Engineering…
mathiq

Can You Succeed in Mathematics? Common Concerns Answered

/
22 Comments
Introduction We often see threads from new members asking whether they are cut out for mathematics, physics, engineering, or other technical fields.…
personaltheories1

Why Won’t You Look at My New Theory?

/
290 Comments
In any forum where science is discussed, there will always be people who have a great new personal theory and can't understand why no one else is interested…
highschoolmath

College-Level Math Options for High School Students

/
21 Comments
This insight is written for high school students who don't feel very challenged by their high school courses or who want a taste of what university mathematics…
virtualparticlesmyths

Top Misconceptions about Virtual Particles

/
1 Comment
Introduction This Insight Article is a sequel to the Insight Article ''The Physics of Virtual Particles'', which contains an exposition of definitions…
acpoweranalysisp3

Power Grid Cyber Resilience: Risks and Defenses Explained

/
9 Comments
Click for Series ListPart 1: The Basics Part 2: Network Analysis Part 3: Cyber ResilienceIntroduction Power system security is much too big…
nightsky

Why Night-Sky Stars Often Appear Brighter Than the Sun

/
3 Comments
Are most visible stars brighter than the Sun? A couple of weeks back, a thread was started about whether or not most of the stars we see at night are…
encryption

Is It Possible to Design an Unbreakable Cipher?

/
10 Comments
Is it possible to design an unbreakable cipher? Do methods of encryption exist that guarantee privacy from even the most capable and highly-resourced…
quantumentanglement

How the EPR Paradox Fits Between Entanglement and Nonlocality

/
11 Comments
If we violate a Bell inequality, we know we have quantum entanglement, and it used to be thought that these two notions were one and the same.At least,…
acpoweranalysisp2

Intro to AC Power Analysis: Network Analysis

/
26 Comments
 Let me use the terms “power grid” and “network” interchangeably. What is the Power Grid Required to Do?Deliver energy to customers,…
Newton-Cartan-Theory

Newton-Cartan Theory: Newtonian Gravity as Curved Spacetime

/
12 Comments
Newtonian limit and spacetime curvature It's a calculation that you can find in any textbook on General Relativity (GR): when gravity is weak and does…
virtualparticles

Learn the Physics of Virtual Particles in Quantum Mechanics

/
33 Comments
In discussions on the internet (including a number of Wikipedia pages) and in books and articles for non-experts in particle physics, there is considerable…
universeexpand

Big Bang Models: Expansion, Dark Energy, Inflation

/
29 Comments
Big Bang and expansion (no inflation, no dark energy) Empty-universe solution (FLRW) Einstein's General Relativity allows a solution (the FLRW metric)…
scienceprojects

Guide to Successful Science Fair Projects & Mentoring

/
1 Comment
Abstract Over the past few years, the author and his wife have served as teachers, qualified scientists, mentors, and/or parents on dozens of science projects…
acpoweranalysisp1

Intro to AC Power Analysis: Learn System Basics

/
13 Comments
 Instantaneous PowerThe formula for power is: ##P=V\cdot I## , power= voltage*current.   We call that, instantaneous power.   Even…
carlo

Carlo Rovelli Interview: Insights on Physics & Life

/
6 Comments
Please give us a bit of background on your life and professional experience Life: Born and educated in Italy; I then spent 10 years in the United States…
scavengerhunt

PF Insights Scavenger Hunt

/
9 Comments
Have you been keeping up with reading each PF Insight? Let's find out! Every question's answer can be found in a PF Insight. Hint links are included if…
johnbaezp2

John Baez Interview — Math, Climate & the Anthropocene

/
13 Comments
Read part 1 of this interview. Tell us about your experience with past projects like "This Week's Finds in Mathematical Physics". I was hired by U.C.…
johnbaezp1

John Baez Interview — Mathematical Physicist & Networks

/
20 Comments
We are proud to introduce you to Mathematical Physicist and PF member John Baez! Give us some background on yourself I'm interested in all kinds of mathematics…
cipher

The Monographic Substitution Cipher: From Julius Caesar to the KGB

/
7 Comments
A monographic substitution cipher works by replacing individual characters of plaintext with corresponding characters of ciphertext. It is perhaps the…
garrettlisi

Garrett Lisi: E8 Theory, Surfing & Physics Interview

/
4 Comments
We are pleased to introduce Theoretical Physicist and Physics Forums member Garrett Lisi, an experimental surfer who developed a distinctive approach to…
geometry1

Pure Geometry Study Guide: Books & Roadmap for Students

/
3 Comments
Introduction to Pure Geometry Why study pure geometry? Geometry is one of the oldest parts of mathematics. It has been studied and advanced by the greatest…
pet_scan

How PET Scans Work: Positrons in Medical Imaging Explained

/
7 Comments
Practical applications of fundamental theoretical physics Positron Emission Tomography (PET) Less often, people ask about potential applications of anti-matter.…
mathanalysis1

Self-Study Analysis Roadmap: From Proofs to Manifolds

/
26 Comments
Introduction This is a sequel to my posts on self-studying mathematics. I have already given a very detailed road map on how to study high school mathematics…
physics_vectors

Frequently Made Errors in Vectors – Elementary Use

/
2 Comments
 A vector has magnitude and direction. Pictorially, a vector can be imagined as a location in n-dimensional space relative to some fixed origin. …
berkeman

Interview with an Electrical Engineer: Berkeman

/
18 Comments
Can you give us a brief history of berkeman? I grew up in a military family (my dad was an Army colonel), so we moved all over the place when I was a…
tcp_ip

TCP/IP Protocols for Plug-and-Play Industrial Automation

/
10 Comments
Background: Ethernet vs Fieldbus in the 1990s The 1990s saw the widespread adoption of network technology on two fronts: Office automation and factory…
Ethernettime

Network Time Synchronization Guide: NTP vs IEEE 1588

/
18 Comments
Overview of Synchronization Needs Now and then you come across measurement problems that are tightly associated with the notion of synchronicity, meaning…
qminterpret

QM Interpretations: Semantics, Meaning & Probability

/
12 Comments
Introduction Many times, we encounter ourselves tangled in some Quantum Mechanics (QM) interpretations debates. Unfortunately, the standard of discussion…
chemist

Interview with a Chemist: Borek

/
37 Comments
Give us a brief history of Borek Note: text below is of a negative pedagogical value, don't read it if you are younger than 20, attending school or still…
spacetime6

Struggles with the Continuum – Freeman Dyson and QED

/
24 Comments
 Last time I sketched how physicists use quantum electrodynamics, or 'QED', to compute answers to physics problems as power series in the…
lightcone7_3

LightCone 8 Tutorial Part III – How Things are Computed

/
1 Comment
In Part I and Part II of this mini-series, we have briefly discussed the basic user interface and the use of charts to depict the LCDM cosmological model.…
universeexpansion

At What Velocity Does the Universe Expand and Can It Be Faster than Light?

Neither of these questions makes sense in the form in which it was posed. To see why, let's start by thinking about how we know the universe is expanding.The…
Mitochondria

When did Mitochondria Evolve?

/
7 Comments
Eukaryote evolution and cellular complexity Historically, life is categorized into three broad domains: bacteria, archaea, and eukaryota.  Whereas bacteria…
zapperz

Interview with a Physicist: ZapperZ

/
22 Comments
This is a new Interview category for Insights. While I line up some great new interviews I’ll be migrating some previous mentor interviews. ZapperZ…
islingmodel

High Temperature and Low Duality for the Ising Model on an Infinite Regular Tree

/
1 Comment
Today I felt like sharing a computation that I thought was interesting. I'm sure I'm not the first to explore the Ising-Ising duality in this context,…
secondaryforces

Introduction to the Secondary Forces in Physics

/
7 Comments
Many are familiar with the "fundamental forces" of nature: gravity, electromagnetism, and the strong and weak nuclear forces. Three of these tend to bind…
scientistquotes

Which Scientist Said What Quiz

/
36 Comments
Think you know your famous science quotes? Let's find out in this 12 question quiz. There are no prizes, so it doesn't pay to Google. Try your best on…
andyresnick

Interview with a Biophysicist: Andy Resnick

/
11 Comments
Let me introduce you to Andy Resnick, a long time Science Advisor at Physics Forums.Tell us a little bit about how you developed an interest in biophysics …
complexmath

Hear the Case for Learning Complex Math

/
15 Comments
Resistance to complex math seems to never die out.  I see it frequently in PF posts.  Often it takes the form of challenges rather than questions. …
spacetime5

Struggles with the Continuum: Quantum Electrodynamics

/
10 Comments
  Quantum field theory is the best method we have for describing particles and forces in a way that takes both quantum mechanics…
complexexponents

Understanding Exponents: Real, Irrational & Imaginary

/
5 Comments
Introduction This is how both you and I learned exponents back in elementary school: 53 = 5 × 5 × 5 Using repeated multiplication you can raise anything…
darkmatter

Knut Lundmark and the Forgotten Dark Matter Discovery

/
16 Comments
Background: textbook accounts Look in your standard cosmology textbook (in my case, the closest one at hand is Barbara Ryden's Introduction to Cosmology)…
docal

Interview with a Physicist: Doc Al

/
16 Comments
This is a new Interview category for Insights. While I line up some great new interviews I’ll be migrating some previous mentor interviews. Doc Al is…
computermath

Why Can’t My Computer Do Simple Arithmetic?

/
46 Comments
The first computer I owned was an Apple IIe computer, with a CPU that ran at slightly over 1 Megahertz (MHz), and with 64 Kilobytes (KB) of RAM, together…
russwatters

Russ Watters: Astrophotography & HVAC Engineer Mentor

/
21 Comments
This is a new Interview category for Insights. While I line up some great new interviews, I'll be migrating some previous mentor interviews. About Russ…
lightcone72

LightCone8 Tutorial Part II – Charts

/
3 Comments
Part I dealt with the basic user interface of LightCone8. This part of the tutorial is about potentially useful cosmological insights to be gained from…
spacetime4

Struggles with the Continuum – Relativity and Quantum

/
5 Comments
 In this series, we're looking at mathematical problems that arise in physics due to treating spacetime as a continuum---basically, problems…
quasarredshift

Evidence that Galaxy and Quasar Redshifts Are Cosmological

/
0 Comments
How do we know that the redshifts of galaxies and quasars are cosmological and not "intrinsic"? Evidence for the cosmological interpretation Historical…
Eigenvectors

What Are Eigenvectors and Eigenvalues in Math?

/
37 Comments
Two important concepts in Linear Algebra are eigenvectors and eigenvalues for a linear transformation that is represented by a square matrix. Besides…
lightcone7

LightCone8 Tutorial Part I

/
2 Comments
LightCone 8 is a versatile tabulating/charting cosmological calculator, useful for understanding the expansion history of the universe (and even some future…
speedlightframe

Why Is the Speed of Light the Same in All Frames of Reference?

Why ask "why" about physical laws? The first thing to worry about here is that when you ask someone for a satisfying answer to a "why" question, you…
complexnumbers

Things Which Can Go Wrong with Complex Numbers

/
3 Comments
At the first sight, there are many paradoxes in complex number theory. Here are some nice examples of things that don't seem to work:Example A [itex]-1=i^2=\sqrt{-1}\cdot\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1[/itex]Example…
theneedforinfinity

Why the Need of Infinity in Physics

/
31 Comments
Infinity in Mathematics and Physics It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics, probably in a finite…
complexscience

Thoughts on the Complexity of Modern Science

/
122 Comments
Science today is incredibly complex. Anyone who ever did scientific research knows that, but many in the general public seem to be completely unaware of…
MillenniumPrize

Millennium Problems: Poincaré, P vs NP, Riemann

/
13 Comments
IntroductionWhat this Insight Covers In this Insight, I will go over the background information for the Millennium Prize problems and briefly describe…
rainbows

Exploration into the Physics of Rainbows

/
88 Comments
For several years, I have been contemplating this beautiful picture by photographer Brian McPhee. I have a personal interest in the photograph because…
micro2

Peano Axioms: Construction of Natural Numbers and Properties

/
0 Comments
Bloch Chapter 1.2The Peano system in Bloch has a special element ##1\in \mathbb{N}##. The intuitive idea here is that ##\mathbb{N} = \{1,2,3,...\}##.…
partialfractions

Introduction to Partial Fractions Decomposition

/
3 Comments
Partial fractions decomposition is an algebraic technique that can be used to decompose (break down) a product of rational expressions into a sum…
micro3

Set-Theoretic Foundations of Numbers and Functions

/
0 Comments
Set-Theoretic Foundations of Mathematics It is important to realize that in standard mathematics we attempt to characterize everything in terms of sets.…
earthgravity

Learn All About Earth’s Gravity

/
18 Comments
Introduction Earth's gravitational field at the surface is approximately 9.8 Newtons/kilogram or 9.8 meters/second/second. But how does that change due…
earlyuniverse

Was the Early Universe in a Disordered State?

No. The second law of thermodynamics says that entropy can only increase, so if the early universe had been in a state of maximum entropy, then the cosmos…