# The maximum range of a missile_the earth is not flat

1. Nov 4, 2005

### 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!

2. Nov 4, 2005

### Integral

Staff Emeritus
When the range of a projectile is such that it "misses" the earth due to the earths curvature the projectile will go into orbit.

3. Nov 4, 2005

### myleo727

The problem is, it does go into an orbit. I think it's an ellipse....

4. Nov 4, 2005

### myleo727

sorry. a part of the ellipse. It'll fall back to the earth.

5. Nov 4, 2005

### SGT

First of all, the 45 degrees value is valid if there is no air resistance. If we take this into account we should use higher angles and this is what is done in artillery.
Secondly, even in the absence of air, the parabolic trajectory is an approximation valid for short range projectiles. In this case we can consider the Earth flat.
For long range artillery, like ICBM, the gravity changes with the altitude of the projectile, so you must consider an elliptic trajectory.

6. Nov 4, 2005

### NateTG

What happens if you shoot the rocket at more than... say... 7 miles per second?

7. Nov 4, 2005

### myleo727

Yeah...I'm now considering the long distance missile, and ignore air-resistance...

8. Nov 4, 2005

### SGT

Then you should consider the change of gravity with altitude (distance to the center of the Earth). The resulting equation of movement is an elliptic trajectory. If you shoot your missile horizontally and with enough velocity, you will have an elliptical orbit whose perigee is at the launch site (supposing no obstacles in the way), so you have a 40000 km range.

9. Nov 4, 2005

### myleo727

hmm. I think I just solved it. Thank you!