I want to compile a set of equations to allow shooters without access to 6DoF ballistics simulations to understand the error introduced by all known external ballistic effects, and relate them to easily obtained data. See examples below.
Links to references, corrections, observations, and anything else that could help would be greatly appreciated. If anyone wants to work on this offline please PM me.
A great starting reference is Bryan Litz's Maximum Effective Range of Small Arms.
Here are the known exterior ballistic effects applicable to small arms, and examples (some guesses) of the sorts of relationships that would be useful:
1. Gravity – affects drop. Proportional to time-of-flight squared. Specifically, drop from a horizontal shot = g t2 / 2 where t is time-of-flight. Force of gravity g averages 32.2 ft/s2. g varies with latitude by up to .5%, and altitude on land by up to .28%. (Ref http://en.wikipedia.org/wiki/Gravity_of_Earth)
2. Muzzle Velocity – affects drop, primarily via time-of-flight, secondarily via drag effects on time-of-flight that vary with velocity. To first order, time-of-flight varies inversely and linearly with velocity. For example, a 1% reduction in velocity will increase time-of-flight by 1%, and therefore increase drop by 2.01%. Second-order effects cannot be generally estimated(?).
3. Air Density (includes temperature, pressure, and humidity) – affects drop, via drag. An increase in air density increases drag and therefore increases drop. Air density = c * p / T, where c is a constant, p is absolute pressure, and T is absolute temperature. Standard density is .002378 slugs/ft3 and will vary up to 30% across normal surface weather conditions on earth. (Can we specify a relationship between air density and drop that holds across all drag models?)
4. Cross-Wind – affects windage, primarily via translation and hence proportionally to time-of-flight and wind speed. I.e., doubling cross-wind will double the windage shift.
Affects drop via aerodynamic jump. (What is the relationship to windspeed here?)
5. Head-Wind – affects drop, primarily via translation and hence proportionally to time-of-flight and wind speed. (Affects windage?)
6. Firing angle – affects drop. (How?)
7. Spin drift – affects windage in the direction of bullet spin, proportionally to drop velocity (which equals g * time-of-flight), and proportionally to spin speed, which is proportional to muzzle velocity and twist rate. So, for example, doubling the twist rate and holding all else equal will double spin drift windage. (To second order is spin drift also, independently, a function of air density?) (Can we put an upper bound on spin-drift windage as a function of time-of-flight and spin rate = twist * muzzle velocity?)
8. Coriolis drift. Varies with latitude and firing direction (“azimuthâ€). Affects both windage and drop. Proportional to time-of-flight. (Maximum effect across azimuth and latitude?)
9. Drag coefficient: Varies proportionally to sectional density. I.e., make a bullet out of a material 50% denser and, all else held equal, drag coefficient increases 50%. (Effect on time-of-flight depends on drag model?)
So we see that the shooter can estimate the effects of any unknown ballistic phenomenon using the following input data, and therefore can also estimate his worst-case error based on failure to account for or properly measure any of the following:
Links to references, corrections, observations, and anything else that could help would be greatly appreciated. If anyone wants to work on this offline please PM me.
A great starting reference is Bryan Litz's Maximum Effective Range of Small Arms.
Here are the known exterior ballistic effects applicable to small arms, and examples (some guesses) of the sorts of relationships that would be useful:
1. Gravity – affects drop. Proportional to time-of-flight squared. Specifically, drop from a horizontal shot = g t2 / 2 where t is time-of-flight. Force of gravity g averages 32.2 ft/s2. g varies with latitude by up to .5%, and altitude on land by up to .28%. (Ref http://en.wikipedia.org/wiki/Gravity_of_Earth)
2. Muzzle Velocity – affects drop, primarily via time-of-flight, secondarily via drag effects on time-of-flight that vary with velocity. To first order, time-of-flight varies inversely and linearly with velocity. For example, a 1% reduction in velocity will increase time-of-flight by 1%, and therefore increase drop by 2.01%. Second-order effects cannot be generally estimated(?).
3. Air Density (includes temperature, pressure, and humidity) – affects drop, via drag. An increase in air density increases drag and therefore increases drop. Air density = c * p / T, where c is a constant, p is absolute pressure, and T is absolute temperature. Standard density is .002378 slugs/ft3 and will vary up to 30% across normal surface weather conditions on earth. (Can we specify a relationship between air density and drop that holds across all drag models?)
4. Cross-Wind – affects windage, primarily via translation and hence proportionally to time-of-flight and wind speed. I.e., doubling cross-wind will double the windage shift.
Affects drop via aerodynamic jump. (What is the relationship to windspeed here?)
5. Head-Wind – affects drop, primarily via translation and hence proportionally to time-of-flight and wind speed. (Affects windage?)
6. Firing angle – affects drop. (How?)
7. Spin drift – affects windage in the direction of bullet spin, proportionally to drop velocity (which equals g * time-of-flight), and proportionally to spin speed, which is proportional to muzzle velocity and twist rate. So, for example, doubling the twist rate and holding all else equal will double spin drift windage. (To second order is spin drift also, independently, a function of air density?) (Can we put an upper bound on spin-drift windage as a function of time-of-flight and spin rate = twist * muzzle velocity?)
8. Coriolis drift. Varies with latitude and firing direction (“azimuthâ€). Affects both windage and drop. Proportional to time-of-flight. (Maximum effect across azimuth and latitude?)
9. Drag coefficient: Varies proportionally to sectional density. I.e., make a bullet out of a material 50% denser and, all else held equal, drag coefficient increases 50%. (Effect on time-of-flight depends on drag model?)
So we see that the shooter can estimate the effects of any unknown ballistic phenomenon using the following input data, and therefore can also estimate his worst-case error based on failure to account for or properly measure any of the following:
- Drag model – produces time of flight
- Drag coefficient
- Muzzle velocity
- Wind speed and direction
- Air density
- Firing angle
- Latitude
- Azimuth
- Gravitation force
