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Ballistics

December 19, 2016

Digital computers were not the first electronic computers. There was a span of several decades in which analog computers were used to solve come common problems. It's fairly easy to implement the mathematical functions of integration and differentiation using operational amplifiers, the earliest of which were built using vacuum tubes. Notable electrical engineer, Robert Pease (1940-2011), who designed some early operational amplifiers while at George A. Philbrick Researches,[1] has written many articles about their history and design.[2-4]

Operational amplifier integratorAn operational amplifier integrator.

In operation, the capacitor is discharged to make the output voltage zero at time zero. Having the same value resistor in the non-inverting input eliminates drifting caused by input bias currents.

(Created using Inkscape.)

Since I'm a member of the baby boomer generation, my physics education included laboratory exercises in analog computation. The one exercise that I remember most is calculating the trajectory of a projectile, the results of the calculation being displayed on an x-y chart recorder. The interesting part of that exercise was seeing the difference in trajectory between a projectile launched on Jupiter as compared with Earth.

The trajectory of a projectile, idealized without atmospheric drag, Coriolis force, etc., has been known since the time of Galileo, and it is derived by a simple addition of the motion arising from the projectile velocity and that of the fall of the projectile arising from gravity,

 trajectory equation

where (x,y) are the points that define the trajectory, θ is the launch angle from the horizontal, v0 is the initial projectile velocity, vh is the horizontal component (x component) of the projectile velocity (vh = v0 cos θ), and g is the gravitational acceleration, 9.80665 m/sec2. An example trajectory, as generated by a simple C language program (source code here), is shown below,

Trajectory exampleAn ideal trajectory without air drag and other confounding influences.

Maximum range is achieved when the projectile is launched at 45 degrees.

(Graphed using Gnumeric.)

Isaac Newton used ballistics to elucidate the idea of an orbit of one body around another, such as the Moon's orbiting the Earth. In his 1728 book, "A Treatise of the System of the World," Newton imagined the thought experiment, illustrated below, in which you fired cannonballs at increasing velocities, finally reaching a point at which they fly into orbit around the Earth.[6] He then generalized this to the Moon's motion around the Earth.

Newton's cannon thought experiment
Newton's own illustration of his cannonball experiment. (From Isaac Newton, "A Treatise of the System of the World," F. Fayram, 1728.[6])

The range R of an ideal projectile is given as R = (v2/g)sin(2θ), so you can hit a target at any range up to a maximum determined by the projectile's initial velocity v by just setting the angle. It can be seen by inspection that the maximum range always occurs at an angle of 45 degrees, since the sine of 90° is one. That angle, however, is not the angle at which the length of the path of projectile travel is maximum.

It's easy to calculate the path length of a projectile as a function of angle. You just use the trajectory example presented above and do a piece-wise integration of the path length. My source code for such a calculation can be found here,[6] and the results of this calculation are shown below with the maximum occurring at an angle near 1 radian (57.2958°).

Trajectory path length as a function of launch angleTrajectory path length as a function of launch angle.

(Graphed using Gnumeric.)

Joshua Cooper and Anton Swifton of the Department of Mathematics of the University of South Carolina have published a paper on arXiv in which they derive an equation to give the angle that produces the maximum projectile path length.[7] This angle, which is independent of the initial speed of the projectile and the value of the gravitational acceleration, is the solution of this interesting equation,

 cosecant-hyperbolic cotangent equation

The easiest way to find the solution to an equation such as this is to use one of the many free and open source software computer algebra systems. Cooper and Swifton used SageMath to obtain an angle of about 0.9855 radians (56.47 degrees). I used my favorite method of refined increments (source code here) to obtain the value 0.985514738.

Intersecting curves, csc(x) and coth(csc(x))Intersecting curves, csc(θ) and coth(csc(θ)).

(Graphed using Gnumeric.)

References:

  1. GAP/R, George A. Philbrick Researches Archive.
  2. Application Brief R1, "Practical closed-loop stabilization of solid state operational amplifiers," Philbrick Archive, February 1, 1961.
  3. Bob Pease, "What's All This Transimpedance Amplifier Stuff, Anyhow? (Part 1)" Electronic Design, January 8, 2001.
  4. Bob Pease, "What's All This Julie Stuff, Anyhow?" Electronic Design, May 3, 1999.
  5. Isaac Newton, "A Treatise of the System of the World," F. Fayram, 1728, pp. 5-6.
  6. Thanks to Anton Swifton, who discovered an error in my original program.
  7. Joshua Cooper and Anton Swifton, "Throwing a Ball as Far as Possible, Revisited," arXiv, November 8, 2016.

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