Calculating Angle to fire at

[craigbutcher]

When seeking guidance, look up. There is a very good website with the basics, not only on ballistics, but also on other cool and significant topics. I am really very impressed by this website. I recommend starting with this index page:

http://exploration.grc.nasa.gov/educ...ket/short.html


You will probably want to be sure to have a look at this particular page:

http://exploration.grc.nasa.gov/educ...et/flteqs.html

Our government has borrowed trillions of dollars to fund this website, so you should take advantage of it!


The strictly ballistic part (without air resistance) you can do parametrically, using the Newtonian relation F=m*a, with first term physics/calculus, or algebraically by looking it up. That's a good first approximation and probably 80% right or so.

The drag part is a much harder problem because the drag always shows up as a force vector whose vertical and horizontal components are changing as a function of the magnitude and direction of the overall velocity vector. Drag in the horizontal direction, for instance, will be largest at the top of the projectile's arc. For this reason you cannot treat the components as ordinary differential equations with separable variables as you can using the simple Newtonian equations of motion. If the projectile is dropping straight down there is no horizontal component, and if it is a car there is no vertical component, which is why you can solve for terminal velocity using simple calculus.

You can make some simplifying assumptions about the average horizontal and vertical drag components, and that will get you closer if you pick the right assumptions. However, your best bet is to do prepare a numerical model using the state equations and some sort of calculation software--you can use Excel, or MATLAB, or Mathematica, or Mathcad, or program it in C. In other words, chop the problem up into N parts of the total flight time T (t0,t1,t2...tN), calculate an approximation at each interval, plug that value back in to calculate the conditions at t2, and so forth. The more the N, the more reality will be willing to cooperate with your answer...if the numbers you pick for the drag coefficient and for the area and so on are the right ones.

Oh, I almost forgot. There is also the Magnus effect which adds another wrinkle yet. A symmetrical spinning or rotating projectile moving through a viscous fluid (for example, air) will experience forces due to pressure differences caused by the Bernoulli effect created by the rotation. This is the subject of curveballs and so on. And while we're on that subject, I should probably mention surface roughness, reynolds number, airstreams, laminar flow, turbulent flow, and chaos theory. Not to worry, though. If even the very smartest of us really understood this stuff (and I am not one of those guys), we wouldn't need to build wind tunnels.

Your numerical simulation will get you in the ballpark. Once in, you need to build a pitching machine and do some testing.

Good luck.