ballisticsMP480 (1)

29 May 2006 CSHPM/SCHPM Annual Meeting

1

The Science of Ballistics:Mathematics Serving the Dark Side

William W. (Bill) HackbornUniversity of Alberta, Augustana Campus


2

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].


3

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”.


4

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.


5

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


6

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).


7

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.

Partially analyzed trajectory when f (V) = kV2. [Hall, 1952]

a

xk

k

g

a

x

k

gby 1log2


8

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:

Bernoulli’s 1721 solution [Routh, 1898]:Letting p = tan θ, where θ is the inclination angle, yields

21

222

122 ,,,

nn

vukvgdt

dvvuku

dt

duv

dt

dyu

dt

dx

dppkngau

np

ab

nn 21

/

21 1

p

ab

p

ab

p

abdppugydpugxdpugt

/21

/21

/1 ,,


9

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

kxs e

ak

g

a

x

ka

gbxYy 2

221

42)(


10

The Ballistics Revolution Benjamin Robins wrote New Principles of Gunnery (1642).

Invented ballistics pendulum for measuringmusket ball velocities. [Steele, 1994]

Did foundational work in interior ballistics.

Discovered Robins effect and sound barrier.

Euler translated and added commentary toNew 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].


11

Late 19th Century to World War I Air resistance per unit mass described by

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]:

C

VGVyHVyf

)()(),(

V (ft/s) 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


12

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].


13

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].


14

Other Social Issues The (mis)use of mathematical and human potential:

Time lost, opportunities missed, e.g. Ramanujan. Time, talent wasted on “such a damnable exercise”.

ICBMs, ABMs, and SDI: 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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