External Ballistics: Deriving Equations for Long-Range Shooting

[MechSoup]

After watching the movie Shooter, my mind began to wonder how difficult it really is to make these “long-range” shots like they portrayed in the movie. Could it really be so easy? So I began thinking of how I could derive such formula’s. I, honestly, didn’t know much about the subject, however I thought I would take a stab at it.

My approach was to first look at the 2-Dimensional world where two perspectives would be observed: without air resistance, and with air resistance. The “without air resistance” part was simple projectile motion and did not yield anything of real world value, so I moved on. When it comes to air resistance things become much more difficult, and thus I had to make some basic assumptions about the relationship of the bullet velocity and the dreaded drag force. I read in a Differential Equations book that for small and slow moving objects the drag force was proportional to the objects velocity, where as, a large and fast moving objects drag force was proportional to the square of the objects velocity.

I began to think that the horizontal velocity of a bullet is relatively slow compared to it’s horizontal velocity (after a given amount a time this would be untrue, but I made the assumption that the bullet would hit the target before this happened) so I made an assumption that is displayed on the below pages.

So to get to the point, I derived some equations and have no idea if they sound reasonable. I really have no experience with the matter to make real world sense of this. This is where I need your help. These are my questions:

1) Does it make sense within the assumptions?
2) Is the constant of proportionality the Ballistic Coefficient?
3) To improve the formula, assuming it is valid, would the formula further hold to incorporate such things as, Coriolis Drift, Wind effects, Air density, ect?

I suppose I should mention that this is NOT a homework assignment. And
that throughout the problem wind effects, ect. were not considered.

Thanks.

NOTE: For the vertical motion equation, I flipped the positive and negative directions.

 

[lsume]

I recently spent a fair amount of time trying to write a mathematical expression to describe the trajectory of a projectile fired from a rifle without using the specific ballistic coefficient. My postulate was that using the simple wind load equation was the major contributor to drag. I'm a retired mechanical engineer and drag is fairly complex. I was going to incorporate the shape of the projectile as a coefficient in the calculation. The hurtle for me was writing the proper integral for handling jerk. From Bernoulli V^2=2*g*h and I used .078 pounds per cubic foot for air's density. I wasn't successful in my attempt to describe the ballistics of a given projectile mathematically without using the ballistic coefficient. It was fun while it lasted. I've all but given up in developing my postulate.