Hello Brachistochrone The brachistochrone is the curve that minimizes the time it takes for a frictionless object to slide along to reach from the top to the bottom across some horizontal span, where the object starts at rest. For this article, I want to introduce the brachistochrone1 from the aforementioned definition. The solution to the brachistochrone problem has been well covered and then some, so for the curious I have included articles for one to learn more!...
Visualizing Roots After watching a 2loop video, I decided to write some Julia code to visualize this my self. Here is the Pluto notebook. With that code I made this pretty visualization. This is actually made with Makie’s datashader method where 20th roots where solved for all combinations of coefficients of $1$ or $-1$. The fullsize image can be generated with the Pluto code.
Falling Under Air Drag A classic physics problem is to compute the time $t$ it takes for a particle to fall from a height $h$, under the influence of gravity and air resistance. This problem can, perhaps surprisingly, be solved analytically with only a handful of steps. Let’s assume a particle dropped from rest (initial velocity zero), subject to gravity $g$ and a drag force proportional to $v |v|$. That is, the force can be written as...
War on Julia Playing real war is no fun the card game called War is1. It takes a while to play though so I thought how long would it at take on average. To do this I programmed war on Julia. Here is the julia notebook code/html view. On avrage it takes about $134$ turns but half the games take less than $100$ turns. Half can go one for a long time....
Going around the circle n times So I was thinking about going around the circle some integer number of times 1 or by some rational fraction of the circle. If you go around the circle $n$ times, where $n$ is an integer, you end up where you started 2. If you go a rational fraction of the circle, you do not end up back at the start after one move. To keep the math straightforward, let’s say we move by a rational number, $\rho \equiv a/b \in \mathbb{Q}$, where $a, b \in \mathbb{Z}$, and we move around the circle $\rho\pi$ radians....
Here is just a quick joke post. Please don’t cancel me! If a 1-body problem is trivial, the 2-body problem is solvable, and the 3-body problem is impossible, why is the 0-body problem sad? Because it has no body!
The Infinite Resistor Grid Nerd Snipe I have to admit it. I got nerd sniped. Really, this blog post is about finding the equivalent resistance in question and more! To be transparent, after a good long think and retries I got the answer, so I will try to document/recollect that journey here. First, let’s set up the problem. The Problem Set Up We are given an infinite grid of resistors, each with a resistance of $R$....
Perusing Reddit I found someone asking about a hanging chain. The post in question has a semi-quality video of someone whirling a chain from one end in a circle. The rest of the freely swinging chain forms a particular shape as it rotates around some center axis. In short, this happens because of the centrifugal force from the rotation axis against the binding force of the chain links. I want to take some time to really understand this with physics, of course....
5 Centimeters per Second …they say it’s five centimeters per second. The speed of a falling cherry petal. Five centimeters per second. According to the hit 2007 animated short film, 5 Centimeters per Second 1, this is the speed at which cherry petals fall. Of course, I’m skeptical, and as a physicist I want to model the heck out of this. So, in this article I would like to propose a model for a thin, circular flower petal to be the spherical cow of flower petals....
Part 2 of this article Traveling in an Expanding Universe Traveling throughout the Milky Way Galaxy, one can assume a flat universe. So we can assume only special relativity to calculate interstellar travel times for all observers 1. This time I want to expand our interstellar travel calculation to the universal scale. On the universal scale, the universe is not flat, however, but is expanding. We can approximate this expanding universe as a so-called flat Friedmann–Lemaître–Robertson–Walker spacetime....