# How big does a rock have to be to keep you from jumping off it?

I’m in space today, just looking at different rocks.

A rock the size of Earth–I can’t jump off it because its got enough gravity that it keeps pulling me back.

A rock the size of a minivan (call it 2000 kg)–I can jump off that and not fall back.

Let’s say I’m a 90 kg kind of guy who can muster a 1 meter vertical jump (when truly inspired) at sea level on Earth. How big does a rock have to be in order to have enough gravity to hold me hostage forever? Since the density of the rock matters, let’s assume I’m dealing with granite at about 2.7 g/cubic cm.

To ballpark it, we’ll assume that your muscles can give you the same liftoff speed from the ground on the asteroid as on the Earth. A 1-m jump on Earth’s surface implies a liftoff speed of 4.4 m/s. If your asteroid is spherical, its escape velocity is given by [scribble scribble scribble] v = √(8πGrho*r[sup]2[/sup]/3), where rho is the density of the asteroid and G is Newton’s constant. Setting these equal and running the numbers, I get a radius of about 2.5 km, or a mass of around 2 x 10[sup]14[/sup] kilograms.

The main problem with such a calculation, of course, is that such a small asteroid is unlikely to be spherical. See Deimos, for example, which is several times larger than 2.5 km but is still noticeably non-spherical.

Out of interest, and if anyone is prepared to do the math, if you were to jump off an asteroid of just below that threshold, how long would you be “airbourne”?

OK, you brung this on yourself. Because now I just happen to find myself upon an asteroid weighing in at around 2 x 10[sup]14[/sup] kilograms. But it’s shaped like a hotdog. Am I better off trying to jup off from one of the ends, or from the middle of the great stone weiner?

Assuming that you want to get off, you need to be as far as possible from the center of mass, so the ends are the way to go.

When jumping off a giant hotdog into space, you want to do so from the point as far as possible from it’s centre of mass, in order to minimise it’s gravitational attraction. My gran always told me.

That depends on just how far below the threshold it is. If it’s close enough, your hang time could be arbitrarily long.

And Heinlein mentioned in one of his books (The Rolling Stones) an Olympic athlete literally jumping into orbit around Deimos.

Good point taking the density into account. But should we assume the same liftoff speed? With lower gravity, your liftoff speed might be higher. Other methods to consider:
-same energy as on Earth
-same momentum as on Earth

Using same energy (logic being that the force you exert is constant, distance is the extension of your legs), I calculate you can jump 900J. With a sphere of radius 3600m, energy on the surface is about 900J less than energy at infinity, so that’s the size you can escape from.

Still figuring out how to do the momentum one.

I would think it’s the opposite. A lot of high jumpers seem to use the first hop and the compression it provides to “bounce” for their big jump. On an asteroid you would not have the traction nor gravitational pull to give you that bounce. In fact, you would have to squat very slowly, or draw your feet to a squat position and wait to sink down to the surface.

How high can you raise your body centre of mass(! Which is about waist level.) from a standing start squat?

Well, the utterly useless answer of jumping off an asteroid just below the threshold is that your hang time would be “just below” infinite. For a more specific answer, consider Mars’ smaller moon Deimos, which others have mentioned as an example of one close to the threshold. It’s escape velocity is actually ~25% higher than the quoted number for jumping (4.4 m/s) in this thread. Assuming you could get 4.4 m/s vertical leap, then you would be able to jump 6 miles high, and your total hang time would be about 3.5 hours.

Of course, that’s just one example. Generally speaking, you will be airborne longer if you’re closer to the threshold. However, you can’t just use the threshold as a way to calculate your hang time. For example, Themisto (a moon of Jupiter) has an escape velocity only 10% above the threshold, but a similar hang time as Deimos since it is more dense.

Could god create an asteroid so big…

Is it safe to assume that the jumper’s mass is a negligible element in the calculations? That a trained circus flea would still be able to achieve escape velocity (assuming the jumps are actually more powerful/faster, not just in relation to body size) on a much larger hot dog?

I dunno. Flea has a much greater vertical leap, pound for pound, than flabby old me, but I have considerably more mass and thus greater momentum to combat the asteroid’s gravity.

I don’t think that ‘momentum combats gravity’ is the best way to think of this - it all comes down to a question of velocity. If you have more mass, then your momentum is greater, but so is the force of gravity acting to pull you back. Those factors cancel out.

If your takeoff speed is the same in both cases, then your takeoff energy (K = mv[sup]2[/sup]/2) will be the same in both cases, as will your takeoff momentum (p = mv).

The main way that the lower gravity might affect your takeoff speed is that gravity is actually doing work on you as you push yourself off. Suppose that your “jump” is just crouching down, and then extending your leg muscles until liftoff. Suppose further that in both environments, your legs do the same amount of work. Then your takeoff kinetic energy, as your feet leave the ground, will be

(takeoff energy) = (work done by legs during takeoff) - (work done by gravity during takeoff)

So on planets with higher gravity, your takeoff speed will be slightly less, since you have less kinetic energy when your feet leave the ground.

Hadn’t thought of that. When you jump, you’re pushing your body up with your leg muscles. If you double the gravity but keep the same musculature our 90kg hero will achieve significantly less than a 0.5 meter vertical. In fact I wouldn’t bet on there being any daylight at all under his feet.

Conversely, in a lower gravity environment like our moon, his legs are moving a total body mass of like 33 kg with musculature accustomed to 200 kg. There is certainly an upper limit to how fast the leg muscles can contract, and thus take off speed, but I’ll bet initial speed will be greater. (There has got to be some way to spin this into an “airplane on a treadmill” type trainwreck)

Almost all animals, fleas included, are about equally good at jumping, in the sense that they can all jump to about the same height. I can jump about as high as a flea, and so can a rhinoceros. People talk about fleas jumping to so much times their own height, but that’s misleading: If you assume that a given percentage of an animal’s body mass is devoted to jumping muscles, and assume that a given mass of muscles can release a given amount of energy, then the energy per mass in a jump should be about the same for all animals. The energy per mass depends only on the speed, and the speed determines jump height, so everyone jumps about equally high. Yes, there are some approximations mixed in here, and so there is some real variation in jump height, but it’s about right.

Nothing to add to the very thoughtful discussion but had to pop in to say that this is why I love this place :D.

This gets my vote as the least likely name for an asteroid, ever.

The world is probably a better place for my never having become a world-renowned astonomer.