The Science of Ballistics:
Mathematics Serving the Dark Side
William W. (Bill) Hackborn
University of Alberta, Augustana Campus
29 May 2006
CSHPM/SCHPM Annual Meeting
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Ballistics and its Context

Ballistics (coined by Mersenne, 1644) is physical science,
technology, and a tool of war [Hall, 1952].

Science consists of interior ballistics (inside the barrel)
and exterior ballistics (after leaving the barrel).

Interior ballistics involves chemistry and physics, the
thermodynamics of combustion and an expanding gas.
Exterior ballistics involves the physics of a projectile
moving through a resisting medium.

Tension between science, technology, and gunnery.

Affected by interrelations among scientists, engineers,
industry, the military, and the state [Hall, 1952].
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CSHPM/SCHPM Annual Meeting
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Niccolò Fontana (Tartaglia)

Mathematical fame from priority dispute with
G. Cardano over cubic equation (1547-48).

The New Science (1537) deals with ballistics.

Designed gunner’s quadrant.

Claimed maximum range at 45º.

Aristotelian and medieval baggage
(violent and natural motion, impetus).

Had qualms about improving “such a damnable exercise”.
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CSHPM/SCHPM Annual Meeting
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Galileo

Did experiments on motion, culminating
in law of falling bodies (in a vacuum) and
parabolic path of a projectile (ca. 1609).
Published in Discourses on Two New Sciences (1638).

Professor in Pisa and Venice. Became “mathematician
and philosopher” to Cosimo de Medici in 1611.

Recognized role of air resistance in causing “deformation
in the [parabolic] path of a projectile”, but …

Thought parabolic theory still valid for low-velocity mortar
ballistics, and included range tables in Discourses.
29 May 2006
CSHPM/SCHPM Annual Meeting
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Toricelli

Galileo’s “last and favourite pupil” [Hall, 1952].

Clarified Galileo’s results in Geometrical Works (1644).

Expressed range as r = R sin 2Φ, where R is maximum
range; designed related instrument.

Dealt with cases where target is above/below gun and
where gun is mounted on a fortification or carriage.

Corresponded with G. B. Renieri (1647) on unexpected
point-blank vs. maximum range, etc. [Segre, 1983].
 conflict of theory vs. practice
29 May 2006
CSHPM/SCHPM Annual Meeting
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Huygens

Used period of a pendulum to determine gravitational
acceleration, g = 981 cm/s2 (1664).

Experiments on motion in a resisting medium (1669):



jet of water impinging on one side of a balance scale
block of wood pulled by weighted cord through water
air screens on two wheeled carts, one pulled at twice the speed

Concluded that resisting force at speed V is given by
FR = kV2, analogous to Galileo’s law of falling bodies.

Abandoned attempt to determine trajectory of projectile
subject to this square law of resistance. [Hall, 1952]

Found trajectory of projectile moving in a medium whose
resistance varies as projectile’s velocity (as did Newton).
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CSHPM/SCHPM Annual Meeting
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Newton

Principia (1687) has 40 propositions on motion in resisting
mediums, investigated experimentally and mathematically.

Concluded that resistance associated with fluid density is
FR = kV2, but resistance may have other components too.

Found projectile trajectory when resistance varies as the
projectile’s speed: FR /m = f (V) = kV.
gx g
kx


y   b    2 log 1 

k a k
a 



Partially analyzed trajectory when f (V) = kV2. [Hall, 1952]
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CSHPM/SCHPM Annual Meeting
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Johann Bernoulli

Solved ballistics problem for f (V) = kVn in response to
a challenge from Oxford astronomer John Keill (1719)
[Hall, 1952].

Formulation of the problem:

dx
dy
du
 u,
 v,
 ku u 2  v 2
dt
dt
dt


n 1
2

dv
,
  g  kv u 2  v 2
dt

n 1
2
Bernoulli’s 1721 solution [Routh, 1898]:
Letting p = tan θ, where θ is the inclination angle, yields
u
n
a
n
t   g 1 
 kng
p
b/a
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1
u dp ,
 1  p 
p
2
n1
2
b/a
x   g 1 
dp
p
b/a
u 2 dp ,

y   g 1 
p
b/a
CSHPM/SCHPM Annual Meeting
pu 2 dp
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How Significant is Air Resistance?

Consider a shot-put, terminal velocity 145 m/s [Long &
Weiss, 1999], projected at 170 m/s at launch angle 45º.

Q denotes Quadratic Drag, i.e. f (V) = kV2.

The small inclination approximation [Hackborn, 2005] is
g x
g

2 kx


y  Ys ( x)   b 

1

e

2 2
2ka  a
4k a

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CSHPM/SCHPM Annual Meeting
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The Ballistics Revolution

Benjamin Robins wrote New Principles of Gunnery (1642).

Invented ballistics pendulum for measuring
musket ball velocities. [Steele, 1994]

Did foundational work in interior ballistics.

Discovered Robins effect and sound barrier.

Euler translated and added commentary to
New Principles, at request of Frederick the Great (1745).

Euler analyzed projectile trajectory subject to the square law
of resistance, calculated range tables for one family (1753).

von Graevenitz published more extensive tables (1764);
still sometimes used in World War II [McShane et al, 1953].
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CSHPM/SCHPM Annual Meeting
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Late 19th Century to World War I


Air resistance per unit mass described by f ( y,V )  H ( y) CV G (V )
where H(y) = e-.0003399y, air density ratio at height y feet,
G(V) = kVn-1, Gâvre drag function,
C = m/λd2, the ballistics coefficient,
λ = form factor specific to projectile shape.
Gâvre function (named after French commission) found
experimentally. Mayevski’s version (1883) [Bliss, 1944]:
V (ft/s)
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n
log10 k
0-790
2
-4.33011
790-970
3
-7.22656
970-1230
5
-13.19813
1230-1370
3
-7.01910
1370-1800
2
-3.88074
1800-2600
1.7
-2.90380
2600-3600
1.55
-2.39095
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Late 19th Century to World War I (continued)

The method of small arcs often used for trajectories.

F. Siacci, at Turin Military Academy, developed an
approximate method for low trajectories with small
inclinations, less than about 20º (ca. 1880) [Bliss, 1944].

Siacci’s method adapted for use in U.S. by Col. J. Ingalls,
resulting in Artillery Circular M (1893, 1918), still
sometimes used in World War II [McShane et al, 1953].

Siacci’s method accurate to O(Φ4), launch angle Φ.

Littlewood, 2nd Lt. in RGA, developed anti-aircraft method.
Improved Siacci’s method to O(Φ6) and high trajectories,
accurate to 20 feet in 60000 for Φ = 30 º [Littlewood, 1972].
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CSHPM/SCHPM Annual Meeting
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Roles of Governments and the Military

Extensive testing was done (e.g. Woolwich, Aberdeen).

Governments in England, Prussia, and France soon
included work of Robins, Euler, etc. in military and
university curricula (e.g. École Polytechnique).

Napoléon, a young artillery lieutenant, wrote a 12-page
summary of Robins’ and Euler’s research in 1788.

Ballistics tables/tools used on battlefields [Steele, 1994].

O. Veblen took command of office of experimental ballistics
at new ($73 million) Aberdeen Proving Ground (Jan. 1918).

N. Wiener worked as a computer at Aberdeen, and later
observed that the “the overwhelming majority of significant
American mathematicians … had gone through the
discipline of the Proving Ground” [Grier, 2001].
29 May 2006
CSHPM/SCHPM Annual Meeting
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Other Social Issues

The (mis)use of mathematical and human potential:



ICBMs, ABMs, and SDI:



Time lost, opportunities missed, e.g. Ramanujan.
Time, talent wasted on “such a damnable exercise”.
Government grants in the mathematical sciences.
Resistance to “Star Wars” in the Reagan years.
When Computers Were Human [Grier, 2005]:



Women in the mathematical work force.
Women in university mathematics and related professions.
ENIAC, silicon chips, and computing technology.
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CSHPM/SCHPM Annual Meeting
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