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Why the 7.9 km/s shot never lands
The maths behind the Newton's cannon demo, in four steps
STEP 1
Gravity pulls the cannonball down
The pull gets weaker the further you are from the centre of the Earth.
STEP 2
To go round in a circle, you need a centre-seeking force
Faster or tighter circles need a stronger pull.
STEP 3
Orbit is when gravity supplies exactly that
The cannonball's own mass cancels out: a feather and a cannonball orbit at the same speed.
STEP 4
Solve for the speed
Plug in the Earth: , ,
THE ANSWER
Fire it at 7.9 km/s and the ground curves away exactly as fast as the ball falls. It is always falling, and never lands.