Galileo Was Right, Nearby

School says a thrown ball flies a parabola. Our balls fly ellipses. Good news: nobody is wrong. It depends how far you throw.

Galileo's parabola

Near the ground, we usually pretend two things. Gravity always points straight down, and it never changes strength. Then a ball fired sideways drops ½·g·t² while it flies v·t forward. That's Galileo's parabola.

Our ground is 0.7 below the muzzle, and g is the pull at the muzzle:

g = 1100 / 11²           = 9.0909
t = √(2 · 0.7 / 9.0909)  = 0.3924 s    every sideways shot lands after this long, at any speed

Both pretend-rules are a little wrong, and they're wrong in opposite ways:

1. The real ground curves away. That lets the real ball fly farther.
2. Real "down" keeps turning toward the centre and getting stronger. That brings the real ball down sooner.

At low speed the second one wins. At higher speed the first one wins:

speed 1    real 0.3816    Galileo 0.3924    Galileo 2.84% too far
speed 3    real 1.1922    Galileo 1.1773    Galileo 1.25% short
speed 8    real 4.9078    Galileo 3.1394    Galileo 36% short

The two errors cancel out exactly at speed 2.567.