# How a ballistics program works

## or how to calculate a trajectory chart

by: R White

Up front I should say that this information is based on how The Shootist, version 2.21, works, and there are other ways of obtaining the information contained in a firing table. I should also set up some definitions to keep the confusion down.

A ballistic table generally does not contain information about a particular projectile's trajectory, but rather general information on the flight of projectiles. The table is then used in conjunction with a set of formulas to produce the trajectory information. Generally, a trajectory chart is computed in the following way:

- Find the remaining velocity and time of flight to the range in question.
- Find the drop of the projectile using this information.
- Correct for the zero of the firearm in question.

All other information, such as kinetic energy or energy transfer, may be found using the information from these calculations.

## Part I: Remaining velocity and time of flight

A table and formulea for using it are included in the file Ingall.txt which may be printed out for convenience in use. This table and its formulea are derived from the Ingall's ballistic tables. These tables were founded upon the test firing of some 'standard' projectiles, and the resulting calculations must be corrected with 'ballistic coefficients.' This is simply a measure of the difference in air resistance between the 'standard projectile' used in compiling the table and an ordinary, common, bullet.

Why is a table used instead of a straight formula? Because it was found by many scientist doing ballistics test that a bullet does not lose speed uniformly. The rate of velocity loss is dependent upon a combination of many factors making the formulea for computing velocity loss very difficult and complicated. The main factor is the sound barrier; bullets lose velocity at a different rate above and below it.

## Part II: Find the drop of the bullet

This is done also on a table, but can be accomplished with formulas. The table used in The Shootist is once again included in the file Ingall.txt. It is based upon the premise that all objects fall at a given rate:

Inherent inaccuracies occur, however, in the formula because of wind resistance, which actually works to slow down the fall of the bullet. The action of the air has already been computed into the table. Instructions on the tables usage are also included. An explenation of the formulea involved in calculating the effect of the air resistance can be found in "Exterior Ballistics" by McShane (and others). The table The Shootist uses is one shown in "Hatcher's Notebook," which is derived from the one in "Exterior Ballistics. To show how much difference there is when the air resistance is figured in, the following comparison is made:

100 Yards | 200 Yards | 300 Yards | 400 Yards | |

With | 1.8 | 7.5 | 17.7 | 33.6 |

Without | 1.93 | 7.72 | 19.76 | 37.36 |

This is based on a bullet beginning at 3200 fps with a ballistic coefficient of .44. The 'With' row was calculated using the table provided in Ingall.txt, while the 'Without' was calculated with the formula given above.

## Part III: Correct for the sighting in of the firearm

This involves a bit of trigonometry that will not even feel like trigonometry:

- H is the height of the sight above the bore
- D is the drop in inches of the bullet at sight-in range
- S is the sight in range in inches
- R is the range in question in inches
- I is the drop from the bore at the range in question
- T is the difference in inches from the line of sight

If the range in question is less than the sight in range

If the range in question is more than the sight in range

The sight in range is not covered by either of the two formulea because it is known to be 0.

For example:

A bullet fired from a gun sighted in at 200 yards drops 4.2 inches in its first 100 yards of travel. How many inches is it above the line of sight? The drop from the bore at the sight in range is 7.5 inches (these are calculated from the tables).

- T = (D-I) - ((S-R) x ((D+H)/S)
- T = (7.5-1.8) - ((7200-3600) x ((7.5+1.5)/7200)
- T = 5.7 - (3600 x .00125)
- T = 5.7 - 4.5
- T = 1.2 inches (above the line of sight)

Another:

The same bullet drops 17.7 inches by 300 feet. How far off the line of sight is it now?

- T = (D-I) + ((R-S) x ((D+H)/S)
- T = (7.5-17.7) + ((10800-7200) x ((7.5+1.5)/7200)
- T = -10.2 + (3600 x .00125)
- T = -10.2 + 4.5
- T = -5.7 inches (below the line of sight)

Other Formulas:

- Kinetic Energy = ((Bullet weight/225218) x Velocity )/2
- Energy Transfer = Kinetic Energy x Caliber
- Optimum Game Weight = Velocity x Bullet Weight x 1.5012 x 10
^{-13} - Mach = Velocity/1127
- Miles per hour = (Velocity x 3600)/5280

Wind drift:

- t = time of flight in seconds
- s = wind speed in mph
- a = angle of wind off trajectory path
- v = muzzle velocity in feet per second
- r = range in yards

I know all this is a bit complicated, but with a little practice it becomes rather easy to make the computations necessary. For further information read "Hatcher's Notebook," which covers a lot of other interesting territory besides exterior ballistics. If you have any questions, leave me a message on CompuServe's NRA forum.

Russ White

ID No: 73737,670