Maximum range of a missile accounting for Earth's curvature

[myleo727]

We know if the Earth is considered flat the maximum range is obtained when the missile is launched at an angle of 45 degrees. But what if we take into account the curvature of the earth? what is the maximum arc-length of the range?
Thank you!

 

[SpaceTiger]

We know if the Earth is considered flat the maximum range is obtained when the missile is launched at an angle of 45 degrees. But what if we take into account the curvature of the earth? what is the maximum arc-length of the range?

Unfortunately, compromising the flat surface pretty much forces you to compromise the uniform gravitational field. Also, you'll have to consider the rotation of the Earth -- the Coriolis effect will make it so that your result depends on the direction in which you fire the missile. What do you need this calculation for?