Projectile hit coordinates at the apex of its path

I have a projectile that I would like to pass through 开发者_运维知识库specific coordinates at the apex of its path. I have been using a superb equation that giogadi outlined here, by plugging in the velocity values it produces into chipmunk's cpBodyApplyImpulse function.

The equation has one drawback that I haven't been able to figure out. It only works when the coordinates that I want to hit have a y value higher than the cannon (where my projectile starts). This means that I can't shoot at a downward angle.

Can anybody help me find a suitable equation that works no matter where the target is in relation to the cannon?

As pointed out above, there isn't any way to make the apex be lower than the height of the cannon (without making gravity work backwards). However, it is possible to make the projectile pass through a point below the cannon; the equations are all here. The equation you need to solve is:

angle = arctan((v^2 [+-]sqrt(v^4 - g*(x^2+2*y*v^2)))/g*x)

where you choose a velocity and plug in the x and y positions of the target - assuming the cannon is at (0,0). The [+-] thing means that you can choose either root. If the argument to the square root function is negative (an imaginary root) you need a larger velocity. So, if you are "in range" you have two possible angles for any particular velocity (other than in the maximum range 45 degree case where the two roots should give the same answer).

I suspect one trajectory will tend to 'look' much more sensible than the other, but that's something to play around with once you have something working. You may want to stick with the apex grazing code for the cases where the target is above the cannon.