The Science of Ballistics: Mathematics Serving the Dark Side

1
The Science of BallisticsMathematics Serving
the Dark Side

• William W. (Bill) Hackborn
• University of Alberta, Augustana Campus

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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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Niccolò Fontana (Tartaglia)

• Mathematical fame from priority dispute with
• G. Cardano over cubic equation (1547-48).
• The New Science (1537) deals with ballistics.
• Designed gunners 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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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.

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Toricelli

• Galileos last and favourite pupil Hall,
  1952.
• Clarified Galileos results in Geometrical Works
  (1644).
• Expressed range as r R sin 2F, 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

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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 Galileos 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 projectiles velocity
  (as did Newton).

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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 projectiles speed FR /m f
  (V) kV.
• Partially analyzed trajectory when f (V) kV2.
  Hall, 1952

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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
• Bernoullis 1721 solution Routh, 1898
• Letting p tan ?, where ? is the inclination
  angle, yields

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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

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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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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/?d2, the ballistics coefficient,
• ? form factor specific to projectile
  shape.
• Gâvre function (named after French commission)
  found experimentally. Mayevskis version (1883)
  Bliss, 1944

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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.
• Siaccis 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.
• Siaccis method accurate to O(F4), launch angle
  F.
• Littlewood, 2nd Lt. in RGA, developed
  anti-aircraft method. Improved Siaccis method to
  O(F6) and high trajectories, accurate to 20 feet
  in 60000 for F 30 º Littlewood, 1972.

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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 Eulers 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.

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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.