Math

Absolute Velocity1 Knowing one’s speed is important in daily life. A 60 km/hr car is much more dangerous than a stationary one. However, the physically important source of danger here is the difference in speed. A 60 km/hr car is 60 km/hr faster than you, so any collision (resulting in a large and dangerous acceleration) would feel not so great. If we moved parallel to the car at the same speed, we would find that we could treat it just as a stationary car....

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...

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....

I was playing on my computer and accidentally tried this expression to see what the computer would output: julia> 9007199254740992.0 + 1.0 == 9007199254740992.0 The REPL output was: julia> 9007199254740992.0 + 1.0 == 9007199254740992.0 true This is obviously incorrect mathematically. However, this arbitrary number is the maximum integer that a 64-bit float can store. In Julia, this number can be found using the maxintfloat function with the Float64 type: julia> maxintfloat(Float64) + 1 == maxintfloat(Float64) true Interestingly enough, adding two “fixes” the comparison:...

The amount of apples can be represented by a number $1,2,3\ldots$. The length and time can also be represented by decimal/real numbers $1.3, \pi, 3,\ldots$. The phase of an electric circuit can be represented by complex numbers! In this sense, physics is useful by abstracting away our world into the world of math. Beyond numbers, another mathematical object is used to represent things like velocity and position. This is the humble vector, an object with both magnitude and direction....

Unit Conversions 1 meter = 100 millimeters = 3.281 feet = 39.37 inches. This is an example of unit conversion. Sometimes, unit conversion is used to convert quantities from an unfamiliar unit to a familiar one. Currently, 1 US Dollar is equivalent to 151 Japanese Yen1 2. At times, the change of unit is for clarity, to express quantities of large or small amounts into comprehensible numbers. For instance, instead of several thousand steps, I find the unit of several miles3 more graspable....

I was watching a YouTube video 1. The video was about the speed of light in a medium, and I thought it would be interesting to do some relaxing programming with Julia. Specifically, I would like to simulate some light propagating through media with varying dielectric properties. This post will follow these steps: Derive the equations from Maxwell’s equations in a dielectric. Simulate the wave equations. Explore! From Maxwell’s Equations Here are Maxwell’s equations as known in a dielectric in general3....

Tests are one of those necessary evils in the world. A good test does what the name implies, it tests the knowledge of the person. Usually this is down to asses progress in a class or for self learning. Though sometimes this could go wrong (or right, keep reading). For essay and long form questions, there is a spectrum of right or wrong. In math and physics and science in general, the dreaded word problem can present many logical traps and common misconception traps....