using the accelerated motion formula from the link above:
pos = vi + v*t + .5*a*t^2
you can calculate the tajectory once you know your accelerations (and inital v)

F=ma can calculate the acceleration you need. Supply the power as a force(F),
and devide by mass(m) to get the acceleration. The wind can also be represented as a force.

since the componets of a vector can be broken up, the sin/cos formula above
allows you to split up your power by the angle supplied.
forcex = cos(angle);
forcey = sin(angle);

First, "power" is not appropriate in this situation. I think a better parameter is "kinetic energy". A gun can give a projectile a fixed amount of kinetic energy, so heavier projectiles will travel more slowly than lighter projectiles. The formula for kinetic energy is E = ms2/2, so the initial speed is this:

    s0 = sqrt( 2E/m )    m is the weight (actually mass) of the projectile 

Next, the initial direction of travel, given an angle, is this:

    D0 = [ cos(angle), sin(angle) ] 

Now with the initial speed and initial direction, you have the initial velocity vector:

    V0 = s * D0 = [ s*cos(angle), s*sin(angle) ] 

Finally, ignoring wind and air resistance, the position of the projectile at any time is given by this formula:

    P = P0 + V0t + Agt2/2    P0 is the initial position    Ag is acceleration due to gravity [0, -9.8 ] 

Now, dealing with air resistance and wind makes this much more complicated (if done accurately). You can simplify things by ignoring air resistance and treating wind as horizontal force. Since, f = ma, and thus a = f/m, you can treat wind as a simple acceleration that is included in the equation. The acceleration due to wind is this:

    Aw = [ fw/m, 0 ] 

The final equation looks like this:

    P = P0 + V0t + (Ag + Aw)t2/2 

So, given the energy of the gun (E), the position of the gun (P0), the angle of the gun (angle), the mass of the projectile (m), and the force of the wind (fw), you can calculate the position of the projectile with the above equation.

Quote:Original post by sofakng
I'm confused on that first equation though. s0 = sqrt( 2E/m )
What does E represent?
Also, in this equation: P = P0 + V0t + Agt2/2
I'm assuming P0, V0, and Ag are all vectors right? What does t represent?
Finally, in your last equation, you have Aw did you mean Fw?

E is the kinetic energy output of the gun. s0 is the muzzle velocity. The equation is simply a rearrangment of the standard kinetic energy equation E = mv2/2. This doesn't match reality but it should be good enough.

P0, V0, Ag, and Aw are vectors. t is elapsed time.

Aw is the acceleration due to the wind. It comes from the force of the wind parameter and the equation a = f/m.