### 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

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 differential operator in terms of one variable into a series of differential operators in terms of othe…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…