I was trying to figure out the best angle to shoot a cannon, with air resistance in the differential equation. I came up with ~36.5 degrees, is that right?
wanna explain how u got that one. It shouldnt depend on anything except gravity mabye and the air resistance possibly. (assuming we are talking about only 1 type of cannon ball)
Well of course you would have to consider the amount of air resistance, but then of course there's the weight of the cannon ball (those are pretty heavy). Most cannons can fire cannon balls pretty far, but to get it to fly the farthest at 45 degrees would take a pretty strong shot. At 36.5 it still would take a little more power than a regular cannon to shoot it farther than 30. I also aimed for 30 because that's the angle that most cannons are aimed at anyway.
Im talking about an idealized situation where the cannon balls are the same and the air is exactly the same. Anyone else have an opinion?
What differential equation? And what value are you using for the air resistance? Don't you think we would need to know that?
For realtively low muzzle velocity, it matters whether the distance is measured at the same elevation as the muzzle. For relatively high muzzle velocity, the radius of the planet you are on becomes an issue. With a high enough muzzle velocity, you could put a projectile in orbit using an angle of zero degrees.
The angle is dependant on a number of variables including initial ball velocity, mass of ball and size of ball (cross sectional area for drag calculation). If you wanted to get more accurate, the variation of air density with altitude and variation of coefficient of drag as a function of velocity would need to be added. There is no single value for the angle which is optimal.
From all these responses it should be clear that you need to specify your initial conditions, and specify the resisting force function. It simplifies things if you can assume the initial speed and height of the projectile are the same in all cases. If you get down to the level of detail of accounting for differences in those initial conditions, you hve more work to do.
The altitude is a factor - especially for very high flights. Angles of 60 degrees can be used to take advantage of the low air resistance at high altitudes.