Day: September 2, 2019

Fascinating Frequencies 1

This challenge is the first of a new science challenge series, Fascinating Frequencies. You will need to use physics to solve this challenge.

Some radioactive particles are decaying. They are decaying in different ways, but each type of decay releases the same amount of energy. If the particle undergoes beta decay, it will release an electron and positron, both traveling at half the speed of light in a vacuum. If it undergoes gamma decay, it will release a photon. The challenge is to find the frequency of this photon.

solution

Peculiar Products

Trigonometry, geometry, ad algebra are required to solve todays challenge.

Pi notation is to Sigma notation as plus is to minus. Sigma notation is the sum of a bunch of stuff, Pi is the product. Pi is also ≈3.1416, but we will note that as π. In Strange Sums, we evaluated an expression with 2 Sigmas. Here will will find the product of the outputs of an equation. Specifically, the equation for the area of a regular polygon. Better go find that one out right now if you don’t know it. We will express this function as A(n, s), where n is the number of sides and s the length of them. Now solve the expression below.

\prod_{k=1}^\infin\Bigg(A\bigg(k,\,\frac1{tan (90\degree)-\frac{360\degree}k}\bigg)\Bigg)
Solution