I'm back to work on Annelids, and I had the need to find the right angle to fire a projectile at a given velocity in order to hit a target. This is surprisingly thorny. Fortunately all the math needed is on this Wikipedia page. But converting to code may be nontrivial, so I thought I'd share what I came up with.
// This file provides utility routines related to ballistics, i.e.,
// projectiles falling under gravity after being launched at some
// anglesToHit: return 0 or 2 possible angles at which a
// projectile could be fired with the given speed in order
// to hit a target at dx,dy relative to the launch point.
// Result is a list of [ang1, ang2] list of options, or 
// if there are no solutions (because the target is simply
// too far away to reach with the given launch speed).
// g is the acceleration (downward) due to gravity, in the
// same units as launchSpeed, dx, and dy.
// Result is in radians.
anglesToHit = function(launchSpeed, dx, dy, g=9.8)
v2 = launchSpeed * launchSpeed
v4 = v2 * v2
term = v4 - g * (g * dx*dx + 2 * dy * v2)
if term < 0 then
// When this term is < 0, the target is too far away.
root = sqrt(term)
gx = g * dx
return [atan(v2 + root, gx), atan(v2 - root, gx)]
if locals == globals then
// Run some simple unit tests.
print "Unit testing ballistics"
assert = function(condition)
if not condition then
print "Unit test failed."
assertApprox = function(actual, expected)
if abs(expected - actual) > 1 then
print "Unit test failure: expected " + expected + ", but got " + actual
// If we fire a projectile at 50 m/s to hit a target
// that is 300m away horizontally and 200m lower,
// we can do that at either 60.6° or -4.3°.
angs = anglesToHit(50, 300, -200)
assertApprox angs * 180/pi, 60.6
assertApprox angs * 180/pi, -4.3
// But if the target is 500m away, we can't hit it at this speed.
angs = anglesToHit(50, 500, -200)
assert not angs
print "Ballistics unit tests passed. Fire at will!"
This is intended to be used as an import module, so after you
import "ballistics" in some other code, then you would call
ballistics.anglesToHit to get the two solutions (or none) to hit the specified target. If you load it and run it by itself, it just does a couple of unit tests and then exits (like the standard import modules in /sys/lib).
Anyway... if someday someone is making a ballistics game, I hope you find this post and that it proves useful. 🙂