## BallisticsDecember 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]
_{0} is the initial projectile velocity, v_{h} is the horizontal component (x component) of the projectile velocity (v_{h} = v_{0} cos θ), and g is the gravitational acceleration, 9.80665 m/sec^{2}. An example trajectory, as generated by a simple C language program (source code here), is shown below,
R of an ideal projectile is given as R = (v, so you can hit a target at any range up to a maximum determined by the projectile's initial velocity ^{2}/g)sin(2θ)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°).
## References:- GAP/R, George A. Philbrick Researches Archive.
- Application Brief R1, "Practical closed-loop stabilization of solid state operational amplifiers," Philbrick Archive, February 1, 1961.
- Bob Pease, "What's All This Transimpedance Amplifier Stuff, Anyhow? (Part 1)" Electronic Design, January 8, 2001.
- Bob Pease, "What's All This Julie Stuff, Anyhow?" Electronic Design, May 3, 1999.
- Isaac Newton, "A Treatise of the System of the World," F. Fayram, 1728, pp. 5-6.
- Thanks to Anton Swifton, who discovered an error in my original program.
- Joshua Cooper and Anton Swifton, "Throwing a Ball as Far as Possible, Revisited," arXiv, November 8, 2016.
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