The Science of Ballistics: Mathematics Serving the Dark Side

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The Science of Ballistics: Mathematics Serving the Dark Side. William W. (Bill) Hackborn University of Alberta, Augustana Campus. Ballistics and its Context. Ballistics (coined by Mersenne, 1644) is physical science, technology, and a tool of war [Hall, 1952].
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The Science of Ballistics:Mathematics Serving the Dark SideWilliam W. (Bill) HackbornUniversity of Alberta, Augustana CampusCSHPM/SCHPM Annual MeetingBallistics 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].
  • CSHPM/SCHPM Annual MeetingNiccolò 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”.
  • CSHPM/SCHPM Annual MeetingGalileo
  • Did experiments on motion, culminating
  • in law of falling bodies (in a vacuum) andparabolic 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.
  • CSHPM/SCHPM Annual MeetingToricelli
  • 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. practiceCSHPM/SCHPM Annual MeetingHuygens
  • 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).
  • CSHPM/SCHPM Annual MeetingNewton
  • 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 whenf (V) = kV2. [Hall, 1952]
  • CSHPM/SCHPM Annual MeetingJohann 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, yieldsCSHPM/SCHPM Annual MeetingHow 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
  • CSHPM/SCHPM Annual MeetingThe 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].CSHPM/SCHPM Annual MeetingLate 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]:
  • CSHPM/SCHPM Annual MeetingLate 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].
  • CSHPM/SCHPM Annual MeetingRoles 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].
  • CSHPM/SCHPM Annual MeetingOther 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.
  • CSHPM/SCHPM Annual Meeting
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