Post #10 "It is a relatively simple matter to show that..." (Physics post)

From Gurnett and Bhattacharjee (hereafter known as G&B): "It is a relatively simple matter to show that..."

This is a fairly common phrase in physics textbooks, and most of the time a student reads that phrase and immediately laughs because usually it is not 'simple' to show whatever it is they want to be shown. It could be simple if a couple key notes were added in the text, I am going to use the above example from G&B.

"It is a relatively simple matter to show that the root-mean-square velocity is given by (root 3) times (C sub s), where C sub s equals (root (kappa times T sub s) divided by m sub s)." I will include a screen shot of my LaTex file, so don't worry so much about the "math into words" I have decided to add:

First things first, root-mean-square velocity. For this problem it is:

Also from an earlier 'easy to show' in chapter 2 of G&B, we know that the denominator is equal to the density of a particular species of particle, n sub s. So the above equation becomes:

The distribution function, f(v), in this case is the following:

So now you can plug it in to the numerator and solve for the root-mean-square velocity. But wait, there is more math that is not for the faint of heart, (feint of heart?). Cue the Google-ing of "definite integral of exponential." Thank you Wikipedia. Skip to this one:

Where n = 4, and alpha equals m sub s divided by (2 kappa T sub s), and don't forget to carry the constants in front of the v and exponential:

To get a cleaner version of the integral in the general form I direct you to a blog post by a friend and colleague: Conrad's blog, and a YouTube video that I found very helpful.
So plugging stuff into this:

You get this:

and after canceling out and putting it back in the numerator of the root-mean-square velocity equation:

you see "It is a relatively simple matter to show that..."

Ta-Da.

Hope this post was sufficiently physics-y and look forward to next weeks post on my sewing projects, including my first circle skirt, and my first dress. Also, two weeks from now I will have my special effects makeup post, featuring my take on an alien inspired by a deep sea creature.