Coriolis Force Along the Surface of the Earth

[whoareyou]

Homework Statement

I don't want to post the actual question because I want to understand the situation in a general case. Basically, there is a bullet that moves south along the surface of the Earth as in this diagram: http://abyss.uoregon.edu/~js/images/coriolis_effect.gif. You have to find the deflection from the target.

Homework Equations

Newton's Second Law in a Non inertial frame, Coriolis Force

The Attempt at a Solution

I don't have a solution becasue I can't understand what's going on. In my textbook, they set up a "local coordinate system" that moves along the surface of the Earth like this: http://i.imgur.com/Eyhq1WF.png. I want to understand why and how they can do this.

I haven't worked out the actual direction of the deflection, but I assume from the picture above it would be westward. If the coordinate system moves along the Earth, can you write a simple DE like $$\ddot{y}=\text{Coriolis Acceleration in this Direction}$$ ? I don't think you can since the latitude changes as you move south.

 

[FactChecker]

There are two components. Here is the first one:
Imagine you are near the Equator and travel North. When you start, you have a lot of velocity to the East because you are at the Equator. When you go toward the North, you retain that velocity. Everyone else there didn't have as high a velocity because their distance from the axis of rotation is smaller than yours was. So unless something stops you, you will be drifting to the East. If you calculate how fast you will drift to the East, you will have part of the Coriolis effect. This part is proportional to the rate that your distance from the axis of rotation decreases.

The second effect is proportional to your East velocity with respect to the Earth: Your velocity to the East adds to the centrifugal force that you would have if you were stationary on Earth. It's as though the Earth were rotating faster and adds a component of acceleration away from the axis of rotation.