Jumping off the planet

I’ve been idly speculating of late: how small would an asteroid, minor planet or satellite need to be for the average person to be able to jump right off it?

The world record for the high jump at present is 2.43 metres. Ouch. More modestly, I guess I can raise my own centre of gravity by at least a metre if I jump up from a crouching position. So using the old equation:

mgh = ½mv2

makes my takeoff velocity around 4.5 metres per second, before gravity inevitably drags me down.

Now, the escape velocity of any spherical body of mass M and radius r is √(2GM/r) – actually, I think I want to chage this around a bit: the volume of yer sphere is 4πr3/3, so its density is 3M/4πr3, so the escape velocity expressed in terms of the density ρ and the radius r is √(8πGρ/3) * r.

Well. The Earth’s escape velocity is about 11 km per second. This is roughly 2400 times faster than I can jump. Since the Earth’s radius is around 6400 km, I could probably jump off a celestial body of the Earth’s density which had a radius of about 2.7 km or smaller

The Earth’s density is roughly 5.5 grams per cubic centimetre (we live on the densest of the planets in our system). Asteroid densities are reckoned to be more in the 1.5 – 3 grams per cubic centimetre range. So in practice, I could probably jump off yer average asteroid or satellite of the 4-6 km size range. Most of the well known ones are bigger than that. Most of the obscure ones, almost by definition, are smaller.

Indeed, Mars’s satellite Phobos, whose mean radius is 11.1 km, has a numerically similar escape velocity of 11.3 metres per second – as you may vaguely remember fron Arthur C. Clarke’s story, “Hide-and-Seek”.

OK, bedtime now.

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