Re: How can I shoot water to its own height using natural air pressure?

Thank you for your question, I hope I can give you an adequate answer. 
First, the short answer to your question is "no". The equation which 
describes the flow of the water was discovered many, many years ago by 
Bernouli (Bernouli's Equation). This equation is essentially a statement 
of the conservation of energy (actually, conservation of energy per unit 
volume in this case) P+rho*g*z+0.5*rho*v^2 = constant; where P is 
pressure at a particular location, say the bottom of your bucket, rho is 
the density of the fluid (water in your case), z is the height and v is 
the speed of the flow. Conservation of energy might be summarized 
by "there are no free lunches".  One can calculate the pressure at the 
bottom of the bucket in the hose from the density of water (rho) * g * zb 
(height of the bucket) * Area bucket/Area of the exit hole. Basically the 
best one can hope for is to reach the top of the bucket with no velocity, 
pressure = ambient air pressure. Reality further intervenes - friction. 
Frictional forces in the hose can lower the top of the possible flow. So 
the laws of physics conspire against you. I hope this helps, but the very 
best you could hope for is for the column of water to barely reach the 
top of the water level in the bucket with no velocity, and frictional 
forces will keep it below that.