# Gravity Assist: the Planetary Slingshot

How does a gravity assist, or slingshot, work exactly for a spacecraft skimming around a planet? Where does the momentum gain come from? Continue reading Gravity Assist: the Planetary Slingshot

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# Category: Tutorial

# Gravity Assist: the Planetary Slingshot

# Why is glass transparent?

# The Copenhagen interpretation

# Spaces and dimensions

# Quantum mechanics in ten ideas for people on the move

# Complex numbers: an introduction

# Heisenberg’s uncertainty principle

# Quantum entanglement: the EPR paradox and Bell’s Theorem

# Quantum entanglement: non-locality and the state of a two-particle system

# Lab centrifuges and prime numbers

# The double-slit experiment

# The Collatz Conjecture

# Proof that the square root of 2 is irrational

# Einstein’s special relativity in under 6.999 minutes for people on the move

# Rainbows: Alexander’s band

# Just a minute: how do polarized sunglasses work?

# Is microwave oven radiation unhealthy?

# Why, exactly, do glass and liquids refract light?

# Finding the normal force in planar non-uniform circular motion using polar coordinates

# Why do wet clothes dry?

# Deriving the volume of the inside of a sphere using spherical coordinates

# Just a minute: what is a black hole?

# Simple problems on relativistic energy and momentum

# Just a minute: how big is the universe?

# The formula that got Albert Einstein the Nobel Prize and should stop us getting sunburn all the time

# Just a minute: why do large and heavy ships not sink?

# Mirror, mirror, what’s up with the mirror writing?

# Why your coffee does not have tides

# Just a minute: Minus minus and negative times negative

# Energy is neither fundamental nor conserved

# When and why do you multiply probabilities?

# The riddle of birthdays

# Simple problems on relativistic energy and momentum

# What is a spacetime interval?

# Deriving the Lorentz transformations from a rotation of frames of reference about their origin with real time Wick-rotated to imaginary time

# Real eigenvalues and eigenvectors of 3×3 matrices, example 3

# Real eigenvalues and eigenvectors of 3×3 matrices, example 2

# Real eigenvalues and eigenvectors of 3×3 matrices, example 1

How does a gravity assist, or slingshot, work exactly for a spacecraft skimming around a planet? Where does the momentum gain come from? Continue reading Gravity Assist: the Planetary Slingshot

If all things consist of molecules, why is it that some materials are transparent while others aren’t? Why is glass transparent? Continue reading Why is glass transparent?

We will have a look at the Copenhagen interpretation as proposed by Niels Bohr and Werner Heisenberg to explain the stranger things in quantum mechanics. Continue reading The Copenhagen interpretation

What do mathematicians and physicists mean with phrases like higher and/or extra dimensions? Let’s travel through a Universe of spaces and dimensions! Continue reading Spaces and dimensions

If you’re on the move, you don’t have much time. Here are ten important ideas from quantum mechanics you can catch up on. Particles and wave functions! Continue reading Quantum mechanics in ten ideas for people on the move

We will leave the domain of real numbers behind us and start exploring the plane of complex numbers. An introduction to a realm beyond imagination. Continue reading Complex numbers: an introduction

Heisenberg’s uncertainty principle is famous in quantum mechanics. However, it doesn’t have its roots in quantum mechanics. Let’s look at Fourier transform pairs. Continue reading Heisenberg’s uncertainty principle

Einstein didn’t like that quantum entanglement seemed to imply information travelling faster than the speed of light. We discuss Bell’s Theorem. Continue reading Quantum entanglement: the EPR paradox and Bell’s Theorem

In this two-parter, we discuss quantum entanglement, non-locality, and some mathematics. In the next post, we discuss the EPR paradox and Bell’s Theorem. Continue reading Quantum entanglement: non-locality and the state of a two-particle system

Lab centrifuges are crucial in e.g. coronavirus research. It’s vital the test tubes are balanced. There is an easy method to know if that’s possible. Continue reading Lab centrifuges and prime numbers

We describe the famous double-slit experiment, which proved to be fundamental to our current understanding of quantum physics. Continue reading The double-slit experiment

The Collatz Conjecture is probably one of the easiest to understand problems which hasnâ€™t yet been answered in the history of mathematics. Continue reading The Collatz Conjecture

We will provide the proof that the square root of 2 is irrational through a proof of contradiction. We will show no valid ratio of integers exists. Continue reading Proof that the square root of 2 is irrational

Especially for people on the move, we discuss as briefly as possible the intricacies of Einstein’s special relativity. Continue reading Einstein’s special relativity in under 6.999 minutes for people on the move

The band between the primary and the secondary rainbow is darker. The area underneath the primary rainbow is lighter. We explain Alexander’s band. Continue reading Rainbows: Alexander’s band

We briefly discuss how polarized sunglasses work, quantum fields, pilots, and 3D cinema. Continue reading Just a minute: how do polarized sunglasses work?

We discuss what electromagnetic radiation is and why ionising radiation is dangerous. We discuss how a microwave oven heats up food, and vitamins too. Continue reading Is microwave oven radiation unhealthy?

We provide a semi-in-depth look into why glass and liquids bend light. We discuss quantum fields, Maxwell’s equations, and vectors. No calculations. Continue reading Why, exactly, do glass and liquids refract light?

We find an expression for the normal force on a mass which is in planar non-uniform circular motion using polar coordinates. Continue reading Finding the normal force in planar non-uniform circular motion using polar coordinates

We discuss the second law of thermodynamics, the notion of entropy, the statistical nature of the situation, and why wet clothes dry. Continue reading Why do wet clothes dry?

Using a volume integral and spherical coordinates, we derive the formula of the volume of the inside of a sphere, the volume of a ball. Continue reading Deriving the volume of the inside of a sphere using spherical coordinates

What is a black hole? We briefly discuss the Schwarzschild radius. Continue reading Just a minute: what is a black hole?

We will focus on a few simple problems where we will manipulate Einstein’s equations for relativistic energy and momentum. Continue reading Simple problems on relativistic energy and momentum

In case your child asks how big the universe is, this is something you quickly might want to read. Continue reading Just a minute: how big is the universe?

Albert Einstein didn’t win the Nobel Prize with his famous formula from the special theory of relativity. What formula did he win the Prize with then? Continue reading The formula that got Albert Einstein the Nobel Prize and should stop us getting sunburn all the time

Until they do due to a mistake, ships do not sink, not even the large and heavy ones. Now and then, textbooks say this is because of dissimilar density. Though not a a wrong statement, it is also not a fundamental one. While ships may sink to the bottom of the ocean thanks to gravity, they also float thanks to gravity. Continue reading Just a minute: why do large and heavy ships not sink?

Ever wondered why sentences, words, and letters always exclusively seem to have their left and right reversed in the mirror, while they are almost never projected upside down? Probably, because mirrors do something else than you would expect. For starters, mirrors don’t reverse left and right. Continue reading Mirror, mirror, what’s up with the mirror writing?

The moon orbits the earth and its gravity is causing the tides. But why don’t swimming pools have tides? Or a cup of coffee? Human bodies consist of water, mostly. Aren’t they tidally influenced by the moon? If you’re asking all these beautiful questions, then what you thought is causing the tides is probably wrong, and here’s why. Continue reading Why your coffee does not have tides

Minus minus is plus. And negative times negative is positive. Two negatives make a positive. You may have heard or uttered these expressions many times. Even though you will know this already, here you will find an algebraic proof, just for your reference. Requirements: simple algebra from the second year in secondary, high or grammar school. Continue reading Just a minute: Minus minus and negative times negative

Sometimes you may have heard someone say that, ‘in the end, everything is energy. Einstein himself said that mass equals energy, we are energy ourselves, light is energy, and everything in this universe is energy.’ Often, it is represented as the fundamental substance everything is made out of. And energy is conserved. Both statements are incorrect. Continue reading Energy is neither fundamental nor conserved

At high school you may have been taught that, sometimes, you have to multiply probabilities. We briefly discuss when and why you do this. Continue reading When and why do you multiply probabilities?

Probabilities can be hard to grasp. For instance, what are the chances that among a birthday party’s attendants two or more people will have their birthdays on the same day? Probably better than you might expect. Continue reading The riddle of birthdays

We will focus on a couple of simple problems where we will manipulate the equations for relativistic energy en momentum. Continue reading Simple problems on relativistic energy and momentum

Einstein and collaborators taught us that space and time are not fixed quantities. They can stretch and contract. They vary. There is one thing, though, that does not vary. It is the invariance of the spacetime interval. Continue reading What is a spacetime interval?

Well-known for their central role in Einstein’s Special Relativity, the Lorentz transformations are derived from the rotation of two frames of reference in standard configuration while time is taken to be an imaginary unit of spacetime. This is rarely seen in the wild. Not many undergraduate textbooks or online texts show the details of the working. Hence, this article. Continue reading Deriving the Lorentz transformations from a rotation of frames of reference about their origin with real time Wick-rotated to imaginary time

We calculate the eigenvalues and eigenvectors of a 3×3 matrix in real number space. Continue reading Real eigenvalues and eigenvectors of 3×3 matrices, example 3

We calculate the eigenvalues and eigenvectors of a 3×3 matrix in real number space. Continue reading Real eigenvalues and eigenvectors of 3×3 matrices, example 2

We calculate the eigenvalues and eigenvectors of a 3×3 matrix in real number space. Continue reading Real eigenvalues and eigenvectors of 3×3 matrices, example 1