Calculating angle for trajectory with exit position not at 0,0

[xal]

I'd like to calculate the angle to shoot a projectile from a cannon in order to hit a target. However, the position that the projectile will be exiting from will not be at 0, 0 -- it will vary according to the angle of the cannon itself. The projectile will be exiting from the tip of the cannon, but the cannon will be rotated about a different point (both points, as well as the target position, will be on the same plane though).

If the projectile was exiting from 0, 0, then I could use this:
$$\tan \left( \theta \right) ={\frac {{{\it v0}}^{2}+\mbox {{\tt `\&+-`} } \left( \sqrt {{{\it v0}}^{4}-{g}^{2}{x}^{2}-2\,gy{{\it v0}}^{2}} \right) }{gx}}$$

However, in this case, I was considering replacing $$x$$ and $$y$$ with $$x-l\cos \left( \theta \right)$$ and $$y-l\sin \left( \theta \right)$$ respectively, where $$l$$ would be the distance away the tip.

That results in this:
$$\tan \left( \theta \right) ={\frac {{{\it v0}}^{2}+\sqrt {{{\it v0}}^{ 4}-g \left( g \left( x-l\cos \left( \theta \right) \right) ^{2}+2\, \left( y-l\sin \left( \theta \right) \right) {{\it v0}}^{2} \right) }}{g \left( x-l\cos \left( \theta \right) \right) }}$$

Different form:
$$-1/2\,{\frac {g \left( x-l\cos \left( \theta \right) \right) ^{2} \left( \tan \left( \theta \right) \right) ^{2}}{{v}^{2}}}+x\tan \left( \theta \right) -1/2\,{\frac {g \left( x-l\cos \left( \theta \right) \right) ^{2}}{{v}^{2}}}-y+l\sin \left( \theta \right) =0$$

But I am not sure if this is a way to do it, and I am not sure to go about solving for the angle either.

 

[Nate810]

Any help would be greatly appreciated! The above equation can be solved for theta by using the Quadratic Formula. Specifically, you can rearrange the equation to the form ax^2 + bx + c = 0 and then solve for x. Once you have x, you can take its inverse tangent to find the angle theta.

 

[kerimek]

I would first suggest verifying the equations and assumptions used in the calculations. It is important to ensure that the equations accurately represent the physical system being studied.

Assuming the equations are correct, the next step would be to solve for the angle using mathematical methods such as substitution or graphical methods. It may also be helpful to plot the trajectory of the projectile and the position of the cannon to visually understand the problem and potential solutions.

Additionally, it may be useful to consider the effects of air resistance and other external factors on the trajectory of the projectile. These factors could affect the accuracy of the calculated angle and should be taken into account in the analysis.

Overall, the key is to approach the problem systematically and accurately, using appropriate equations and methods to solve for the desired angle.