## Sunday, May 12, 2019

### [xgaqjchh] How deep is your gravity well?

Express the depth of a hole in terms of the velocity needed for a projectile launched straight up from the bottom to reach the top of the hole.  For holes not very deep, we assume gravity remains constant.  We ignore air resistance, even though that is a pretty bad idea at high velocities.  Some useful formulae:

v = sqrt(2*g*h)
h = v^2/(2g)

1 cm = 0.44 m/s
1 cm = 1 mph
0.40 inch = 1 mph
1 inch = 1.6 mph
5.1 cm = 1 m/s
1 ft = 5.47 mph
1 m = 9.91 mph
1 m = 4.43 m/s
1.01 m = 10 mph
3.34 ft = 10 mph
6 ft = 13.4 mph
10 ft = 17.3 mph
4.08 m = 20 mph
13.4 ft = 20 mph
5.1 m = 10 m/s
29.9 ft = 29.9 mph
10 m = 14 m/s
19.6 m = 19.6 m/s (where 19.6 = 2 gravity)
100 ft = 54.7 mph
120 ft = 60 mph
39 m = 100 km/h
334 ft = 100 mph
99 m = 100 mph
100 m = 44 m/s
510 m = 100 m/s
1 km = 140 m/s
1 mile = 397 mph

Some cute numerical coincidences: 1 cm = 1 mph; 1 m = 10 mph; 20 m = 20 m/s; 30 feet = 30 mph.

For the gravity well of the earth itself, the critical speed is escape velocity, 11 kilometers per second or 25000 miles per hour.  This is coincidentally 1 earth-circumference per hour.  Escape velocity "works" even if you are initially traveling tangentially the surface, though ignoring air resistance is then even more stupid.

Incidentally, escape velocity is (only) (sqrt 2) times the orbital velocity, independent of attitude.  So 2x kinetic energy, though we probably need the rocket equation to compare the fuel needed.