On Earth, a typical human being can jump straight up to a height of something between 12 to 24 inches, and then comes back down. On the Moon, one can jump significantly higher (discounting the burden of a couple of hundred pounds of atmospheric gear; assume a sealed chamber) but one still could not leap oneself into orbit.
How small would a body (small moon, asteroid, whatever) have to be for its gravity to be weak enough that the typical human could propel himself into orbit around it? Is it even feasible? In other words, to have low enough gravity that a human could leap into an orbital trajectory, would the body have to be so small that gravity becomes impractically negligible, to the point that one couldn’t effectively “stand” on the body in the first place?
I can’t answer except to add that a theme of some SF is about folks who find themselves adrift after becoming unstuck from small planetoids.
I’m guessing that if you can specify the speed of the jump you can figure the size of the planetoid if you specify the density. They’d all be interrelated. Don’t know how to do the math though. I’d guess that an NBA basketball star could jump clear of a larger planetoid that Hulk Hogan could.
And of course for orbit there would need to be a horizontal component to the jump, as opposed to straight out from the surface.
You can’t “jump” into orbit from the surface of a body, even with horizontal velocity.
The reason is that if you only have a single impulse, you need sufficient orbital speed, but also need to start off from above the surface.
Any trajectory that starts from the surface will intersect the surface again (unless it has sufficient velocity to escape).
Jumping (or firing a cannon) from the surface of a boy can only put you (or a projectile) once around the body to the point of launch, or off on an escape trajectory–but cannot result in a stable orbit.
Now if you use a rocket to both lift you off the surface and then impart a horizontal velocity, you are OK. You are also OK, if you start off above the surface, like Newton’s mountain.
For a 1km diameter iron sphere (vol = 523597024 cubic meters, density = 7874 kg.m^3, mass = 4122802966976 kg) escape velocity is 1.05 meters per second, 2.4 mph.
That’d be easy for a jumper to achieve.
Assume you leap from the highest mountain on the surface. Your orbit will bring you back to the same place, but if the planet has any rotation, the mountain won’t be there anymore. You’ll stay in orbit until your path and the mountain happen to coincide again.
But your main point is perfectly valid. Unless you have some way to alter your trajectory, an orbit will bring you back to where you started.
As has been noted, you can’t jump into orbit, but, assuming just leaving the gravitational well is enough for your purposes, here’s some back-of-the-enveloping:
Energy conservation gives us:
E[sub]pot[/sub] = E[sub]kin[/sub]
with E[sub]pot[/sub] being the gravitational potential:
GmM/r = 1/2 * mv[sup]2[/sup]
using the density rho = M/V and approximating the body as spherical, we can substitute M = rho*V = 4/3 * pi * r[sup]3[/sup] * rho, giving the radius:
r = v * sqrt[3/(8Gpi*rho)]
(if I didn’t fuck up anywhere), and, using 3.5 m/s as an estimate for the speed (roughly corresponding to a jump height of 24 in = 61 cm), we get a radius of roughly 1.98 km (somebody please check my numbers, I always fuck stuff like that up; the density I used was that of the Earth, about 5520 kg/m[sup]3[/sup]).
That asteroid would have a mass of about 1.8 * 10[sup]14[/sup] kg, and you’d be roughly 3200 times lighter than on Earth. Again, if I didn’t fuck up.
ETA: Squink’s link seems to validate my calculations, at least for escape velocity, mass and radius.
This is where I stopped. Wouldn’t the speed be higher than the v[sub]0[/sub] value on earth. With the same force and a smaller gravity (negative acceleration), won’t the net acceleration be higher, giving a higher v[sub]0[/sub]? I kept trying to back-calculate, but kept getting an acceleration of g. :smack:
Yes, a little bit. It’s more or less the difference in gravitational acceleration over the time that the muscles are accelerating the person up – so the time from when the person starts jumping to when the feet leave the ground. I’m guessing for a typical jump this is about 5 hundredths of a second (half of 0.1 seconds), so this means the v[sub]0[/sub] value on earth is at most 0.5 m/s second less than it would be in zero-g. The difference on a planetoid would be less of course.
You’re right, I didn’t really consider an actual human jumping process, rather something that just starts out at a fixed velocity, unphysical as that might be; the problem is to know how long it takes the jumper to reach his final velocity – about 0.1s? If so, I’d estimate about 4.5m/s in a negligible gravity environment, if it’s faster, final speed will be lower (however, we can just assume the 3.5m/s for an average jumper, I guess ;)).
ETA: Beaten to it. As an addendum, it probably wouldn’t make much of a difference whether you’re in actual zero-g or on the hypothetical asteroid, since it’s g’ = GM/r[sup]2[/sup] is something around 0,003 m/s[sup]2[/sup].
I did think about this before I posted the question, I swear; I just forgot to say anything about it. :smack:
What I had in mind, before my apparent oversight-in-haste, was to carry some large rock or other meaningful mass with you when you jump, so you could throw it away at the apogee to angle yourself into a new trajectory, and thus prevent yourself from looping around and smacking back into your takeoff point.
That said, thanks for working out the figures. I’m a little surprised the body is as large as that.
Science fiction fans may recall that Heinlein, in The Rolling Stones, mentioned one Ole Gunderson, an Olympic athlete who successfully jumped around Phobos.
If anyone’s planning on trying this, you’ll get the best results by waiting until you’re at your highest point, and then aiming your fire extinguisher straight back in the direction you came from and emptying it out as fast as possible.
“Best” here, I assume is the most circular, least eccentric orbit? If so, probably should qualify that instruction. Do not point the exhaust down back at your launch point; keep it perpendicular to the center of the body you’re trying to orbit. When you are at the aposis point, that does coincide with the opposite direction in which you’re moving, but assuming you’re on a planetoid big enough to have noticeable gravity, but small enough for this trick to be doable, it’s going to be awfully hard to determine the exact aposis time without fancy instruments that aren’t usually installed on your typical spacesuit.
Some trajectories might have your aposis point right back at the mountain you launched from. You’ll need more than one burn, or at least change the direction of your nozzle during the burn, to get yourself into a circular orbit that misses the mountain. Try this simulation and use 6.84 km/s as your launch velocity. Newton's Mountain
For what it’s worth, my original question was supposed to be about somebody getting himself into orbit using human muscle power alone. That’s why I distinguish between the use of a fire extinguisher or some other rocket-like device and the “throw a rock!” vector-altering method I mentioned above, because it doesn’t require anything more mechanical than human strength. Not that it really matters, of course; this was just one of those random things that bubbled up in my brain while sitting in bumper-to-bumper traffic, and I knew the Big Brains of the SDMB would be able to work out the concrete details. Anyway, thanks, y’all.
You could take a bicycle pump and an inner tube. As long as the pump draws air in from every direction (or better still, sucks you in the direction you want to go) you can pump up the inner tube; then, at the right time, press the valve and get a small push in whatever direction you want. And that’s repeatable.
For that matter, you could do it just by breathing. Use a snorkel when you breathe in (to draw in air from behind you), then blow. You’d probably waste a lot of energy tumbling end-over-end that way, but you’d get some thrust out of it.
If gravity is low enough to jump into orbit, surrounding air is probably going to be pretty scarce.
Chronos – I’m having trouble understanding what you’re saying… If you jump straight up from a (non-rotating) planetoid, then at the highest point fire the jet straight down, you’ll still be in an orbit that intersects the center of the planetoid, right?
Try this simulation and use 6.84 km/s as your launch velocity.