### Projectile Motion: Pushing the Envelope

Think everything that's publishable for say an old classical topic like projectile motions has already been published?  Turns out the old 'lob the projectile at a constant velocity in a constant gravitational field' problem is still producing.  Check out this paper from J. L. Fernandez-Chapou, A. L. Salas-Brito, and C. A. Vargas published in 2004.  It eventually made its way into the American Journal of Physics.  In the paper, the authors show that if you write down the trajectory of a projectile in terms of its launch angle and then solve for the x and y position when the projectile has reached it's maximum height, the solutions will trace out a nice little ellipse like the figure below excerpted from the arxiv version.

References:
1.  Elliptic envelope of parabolic trajectories paper
http://arxiv.org/abs/physics/0402020v1

1.a.  AJP version of the paper
http://scitation.aip.org/content/aapt/journal/ajp/72/8/10.1119/1.1688786

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

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The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differe…

### Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…