You know I like the MythBusters, right? Well, I have been meaning to look at the shooting bullets in the air myth for quite some time. Now is that time. If you didn't catch that particular episode, the MythBusters wanted to see how dangerous it was to shoot a bullet straight up in the air.

I am not going to shoot any guns, or even drop bullets - that is for the MythBusters. What I will do instead is make a numerical calculation of the motion of a bullet shot into the air. Here is what Adam said about the bullets:

- A .30-06 cartridge will go 10,000 feet high and take 58 seconds to come back down
- A 9 mm will go 4000 feet and take 37 seconds to come back down.

Adam was also able to experimentally determine that both the 9 mm and the .30-06 have a terminal speed of about 100 mph. So, that is what I have to work with. Oh - also, they measured how far a 9mm bullet penetrated into the dirt (but they couldn't find the .30-06 ones).

This is actually similar to Hancock throwing a boy. The basic plan is to use a numerical calculation to model the motion of a bullet. After the bullet leaves the gun, it has forces acting on it like this:

I made two force diagrams because the air resistance force is going to be in opposite direction as the motion. This means that moving up bullet will look different than going down. So, this problem seems simple enough - right? I have actually done this before (here is an example of the air resistance on a football). But in this case, there are some other things to consider.

- Does the normal model of air resistance work (being proportional to v
^{2})? - What is the drag coefficient of a bullet?
- What about the density of air? Do I need to take that into account?
- What about the change in gravitational field of the Earth as the bullet moves up?

I don't want to go into the details, but in case you forgot, the numerical calculation works this way:

- Break the motion into tiny little time steps. During these steps, I can pretend (assume) that the force is constant. With a small enough time, this is true enough.
- For each time step: Calculate force
- Calculate change in momentum (assuming constant force)
- Calculate change in position (assuming constant momentum)
- repeat