John P Knox Math 308 Project

The Mathematics of Artillery:

A Study in Modern Ballistics



          While for most artillery users, the art of artillery has come down to inputting variables into a computer program that is essentially a large experimentally derived spread sheet, the science of ballistics has driven mathematics and physics for most of the last millennium and complex theories have evolved around them.  The trouble with ballistics is the complexity of the formulas and the need for constant reevaluation in flight. 





          F coefficient

            n  coefficient

            Gravity and Drag

            Velocity and Drag

            Flight Times


Moving Targets





The study of ballistics began with the advent of the cannon which occurred somewhere around 1327 when Edward III began producing Spingards, an early cannon.  For the next two centuries, the mathematics of ballistics was left to those operating the cannons.  At this point, it was believed that a projectile would essentially travel in the same vector that they were fired and would not deviate from this path more than from the inherent inaccuracies of the device.  Indeed, early in the history of ballistic projectile warfare, cannons were more for show and fear then for strategic value.  As the weapons became more reliable, gunners began to experiment with angles and power compositions in an attempt to better aim their weapons.


Niccolo Franco Tartaglia, an Italian mathematician, developed the science of ballistics.  Tartaglia was determined to discover the ideal firing angle to achieve a maximum range.  He experimented with a variety of cannons and derived the correct answer of 45 degrees but was also aware of many inaccuracies in his data due to external effects (such as drag).  Tartaglia, who had experimented with almost every type of cannon in existence in Europe, had a great deal of data on cannons and so was able to develop the first ballistic firing tables; these tables were instrumental in educating gunners and developing artillery as a precise military tool.


  Range is maximized by a 45 degree firing angle.


          The next major contribution to ballistics came from Galileo Galilei who showed that the acceleration due to gravity is the same for all objects and air drag was the factor that changed their descent velocities.  He was able to determine that ballistic trajectories are parabolic.  He theorized that the velocity of a projectile was related to the drag acting upon the projectile but erroneously assumed that gravity a greater effect that drag.  He lacked the ability to ascertain the effects of drag at high speed and so was unable to measure how air drag retards a projectiles velocity.

The basic ballistic profile


          The ability to measure the velocity of a projectile and therefore study the effects of drag was not conceived till 1730 when Englishman, Benjamin Robbins, invented the Ballistic Pendulum.  The ballistic pendulum is essentially a block of wood suspended from a string.  The projectile is fired horizontally into the block of wood.  The maximum angle and the height to which the block swing allow their speed and momentum to be calculated.  From this data, the blocks momentum and its speed just before impact can be easily determined.  Robbins found both the initial velocity of a projectile and also the change in velocity over considerable distances.  He was able to observe the incredible deceleration of a high speed projectile and found that drag affected a projectile 50 – 100 times more than gravity.

A drag free environment would have an exponential range function.


          Sir Isaac Newton made the most important contributions to ballistics and the study of aerodynamic drag. In Principia, he derived formulas and explained the mechanics of ballistics.  He concluded that the retarding force (drag) that acts on a projectile through air is proportional to the density of air, the cross sectional area of the projectile and approximately the square of its velocity.  This relationship holds fairly well even now for objects well below the speed of sound.


            Leonard Euler tried (with some success) to solve the complex trajectories of ballistic flight.  He developed the Euler method mainly to simplify ballistic equations and integrate the various stages of a flight path into one.  The formulas that follow are along those lines of simplified trajectories.


Modern Ballistic Theory



          Aerodynamic drag involves two forces: that of air pressure drag and that of skin friction.  Air pressure drag is caused by the fact that an object disturbs the air flowing around it, causing the air to separate from the objects surface.  Areas of low pressure form behind the object that result in a pressure drag on the object, essentially pulling the object backwards.  Skin friction drag occurs where the air comes into contact with the surface of the object and is less of a factor than air pressure drag.



The difficulty is to express drag as a function related to the known ballistics equations.


Retardation Coefficient F


Where V is velocity and and A is deceleration due to drag.


F is a constant for a particular projectile and so must be derived and recalculated for each projectile and muzzle velocity (the velocity at which a projectile leaves the artillery.)


F is c times the small range interval in which a projectile loses 1/c of its remaining speed due to drag.



F = 2700 (which means for every 27 meters, the projectile loses 1% of its velocity)

Say a projectile is fired at 1000 m/s, after traveling 27 meters, it will have had its velocity reduced to 0.99 (1000) = 990 meters and so on such that:


 Where r is the distance traveled.


F can be calculated by comparing two velocities at two points:


Steepness Exponent n


F is modeled proportional to V raised to a power n such that: 


To find n, values of F are needed at two velocities and two ranges.  One can then use:

 to find n.


We see two sets of experimentally determined retardation coefficients based on two different caliber bullets.  The n values are approximated.

The peaks reflect when the projectiles pass through the sound barrier where they attain their maximum drag.



Gravity versus Drag

Gravity can be ignored when compared with drag when velocity is considered.


Eg:  On a horizontal trajectory over 300 yards, gravity causes 10 fps of downward velocity in relation to the original vector while drag reduces the horizontal velocity by 630 fps (considering an average retardation coefficient.) So using the Pythagorean theorem:


Hence gravity can be essentially ignored.


Drag Changing as Velocity Decreases

          Drag reduces as the velocity decreases so the easiest way to calculate this change is to recalculate the drag over several intervals.  In general, the greatest percentage of velocity is lost over a short period of time but a large distance and then the change in distance over time reverses (longer time at a given range of velocity but shorter distance.)




Comparing Flight Times of a Projectile using either a Series of Steps or using one Interval


          The formula for computing flight time over a distance using drag is:

Where  are the initial retardation coefficient and the initial velocity.


This was the area Euler made his greatest developments in the field of ballistics.

With  set at 3000 fps,  set at 4000, n starting at 0.5 and x = 1200 feet.  With n remaining the same, T = 0.46194 seconds, with three changes of n as the velocity lowers (n =1/3, n =1/5 and n =0), T only changes by 0.00013 seconds or an  0.028% difference.


The Wind                                                          


The wind effects a projectile with the following relation:

 where  is the velocity of the wind.


Shows the increase in deflection (cm) as range increases (m)


Knowing the deflection that the wind will cause, one can then use the angle at which the wind is at to determine where the wind will direct the projectile.


The Wind angle is the direction of the wind with relation to the original ballistic path.




 By rotating the ballistic launcher by the angle of deflection                                                                                                                                                          error, just have to alter range slightly to hit target.


Moving Target

The hitting of a moving target is actually simple than one would at first assume as long as the target is on a predictable path with an observable velocity; the key is to fire to lead a target to the correct distance.  This lead is just distance that the target will cover during the time taken for the projectile to reach its target.  In order to do this, one must calculate the flight time (which has already been shown above) and then multiply that by the target’s own velocity.

Eg. Muzzle velocity of 3000 fps, target at 400 yards and moving at 50 km/h or 44 fps (very close, but we can imagine a tank versus tank battle where a moving target would be important,) then the time to target would be T = 0.47 seconds. So: and so would have to fired 20.7 feet in front of the moving target.


Ballistics Today

3d representation using Bezier curves


Today, every gun of any size or shape (whether battleship or air rifle) has had its ballistics analyzed to determine every possible variable.  All this data is then put into vast spread sheets and a ballistics computer breaks the data down into several categories.

-Muzzle Velocity

-Projectile Mass

-Projectile Shape Class (What is it)

-Distance to target

-Wind speed and direction

-Target velocity



The computer then finds the ideal firing sets based on pre-established data and returns that data.


          The science of ballistics has not changed much in the last 300 years from the point of view of the person firing the artillery.  They are still just inputting data into a firing table (though now computerized) and using that data to fire at a target.  The science of ballistics has evolved though as a greater study of the atmospheric conditions and the affects of drag at higher speeds.  The X-1 (the first manned object to surpass the sound barrier) was designed to look like a bullet with control surfaces mainly because it was known that a bullet could exceed the sound barrier but not why.  Modern ballistics is all about designed a projectile that resists drag at its prescribed flight profile (e.g. Supersonic vs. subsonic) rather than getting to the target.



Created by: John P Knox 79314019

            For: Math 308 Final Project: Professor Bill Casselman

Last Edited: December 16th, 2005