[Guest]

Imagine the earth is split into two halves. you are on one half at the very edge of the half.

if you jump off the edge:

how far do you fall before you stop falling?

Which way do you fall? down up sideways?

[Guest]

Just to let you know I don't have an answer. just something i used to think about as a child.

[superprismatic]

A similar problem about jumping into a hole drilled all the way

through the earth is discussed here where the author says the following:

"The traveler accelerates toward the center of the Earth and is

momentarily weightless when passing through the geometric center at

about 7900 m/s or almost 17,700 miles/hr. The traveler would pop up

on the opposite side of the Earth after a little more than 42 minutes.

But unless he or she grabs something to hold on, they will fall back

for a return journey and continue to oscillate with a round-trip

time of 84.5 minutes." Nice stuff, that!

[k-man]

First of all you would not be able to survive the earth splitting into 2 halves especially if you happen to be standing right at the breaking point. However, if we ignore all the "unrelated" aspects of the earth splitting, lava, etc. then it depends on whether the other half is still around or not...

1. Both halves are next to each other as if the earth just split and there is a narrow gap between them and you're jumping into that gap. You will fall all the way to the center of the earth as this is still the center of gravity.

2. The other half is gone and there is only one half left. In essence you are stanging on the "edge" of a hemisphere shaped planet. The center of gravity is inside the hemisphere, so it's like standing on the peak of a mountain. There is no vertical drop, but rather a steep decline, so you can't fall, but if it was snow covered it could be quite a thrill skiing down that slope

EDIT: added the spoiler

Edited November 11, 2010 by k-man

[Guest]

A similar problem about jumping into a hole drilled all the way

through the earth is discussed here where the author says the following:

"The traveler accelerates toward the center of the Earth and is

momentarily weightless when passing through the geometric center at

about 7900 m/s or almost 17,700 miles/hr. The traveler would pop up

on the opposite side of the Earth after a little more than 42 minutes.

But unless he or she grabs something to hold on, they will fall back

for a return journey and continue to oscillate with a round-trip

time of 84.5 minutes." Nice stuff, that!

What about Coriolis? I think you're right, but i'm not sure.

[EventHorizon]

If the two halves just separated by a little bit (and magma didn't fill it in...), then the center of gravity would be empty space. You'd fall through it, and rise back up a fair amount on the other side (but not to the edge due to air resistance). You'd oscillate for a while. The gravity near the center would be near zero, so you will not end up right at the center. It may be like in space... you could just float around.

If the other half of the earth is gone (and magma doesn't move... or else this side of the earth would turn more spherical as things collapsed), then you'd still fall towards the center of gravity of the hemisphere. Because I'm too lazy to figure it out, I googled it and it turns out that it is 3r/8 from the center of the would-be sphere. This means if you were standing at the edge, you'd actually be standing at the angle arctan(3/8) (~35.877 degrees) over the edge instead of standing what used to be straight up. You may even be able to walk over and across the flat side of the hemisphere. It would start off as a steep decline, and get more flat as you get closer to the used-to-be core (since the direction to the center of gravity would get more straight down). I guess it would feel like walking down a giant crater. I think the gravity once you reach the center would be less than when you were at the edge due to more competing forces (gravity from the edge of the hemisphere essentially cancel out, etc).

Then again, you'd probably need to calculate the force of gravity for the specific point, so the angle may not be to the center of gravity of the hemisphere since the matter closer to you pulls more than the further... again, too lazy to figure it out.

[Guest]

What about Coriolis? I think you're right, but i'm not sure.

V X r = 0

[Guest]

The gravitational pull of the hemispherical half-earth would be significantly less (less than half) than earth's gravity since it is half the mass.

[Guest]

The gravitational pull of the hemispherical half-earth would be significantly less (less than half) than earth's gravity since it is half the mass.

true

[phaze]

If you were to remove half of the earth what you have effectively done on a large scale it to create a bowl shaped valley (it might not look like one). The center of mass of the 1/2 earth would change to being the middle of the resulting hemisphere. What you could do in this situation (unlike having a single hole where you would simply oscillate) is run around the edge or rim and build up some speed so that you travel in an elliptical motion around the bowl. Admittedly at this scale unless you have attached rockets to your back the ellipses will be elongated to an extent that nobody would notice any difference from simple oscillation. I figure that if you are daredevil enough to slice the earth in two just to play in the resulting hole without protective gear that procuring the additional rocket backpack will not be an issue.

]

[bonanova]

Imagine the earth is split into two halves. you are on one half at the very edge of the half.

if you jump off the edge:

how far do you fall before you stop falling?

Which way do you fall? down up sideways?

If the halves are still close to each other, there are two possibilities:

Say they're 10' apart - farther than you can jump. Then you'll fall to the center of the earth. That is, to a point 5' distant from points that were the center before the split. That case would be bad. First, you'd overshoot, limited only by wind resistance and get somewhere near China, possibly, then fall back again, etc. until wind resistance brought to to a stop, at the center, where, because of the intense heat, you'd become a cinder.

Now instead, say they're 10" apart. You'd easily clear the gap, and you'd hardly detect the difference from the case that the earth is not split at all. You'd end up much better off in this case than in the previous case.

Now, consider that the halves have been separated by a distance large compared to an earth diameter.

Well, first, this would be a very bad thing for a lot of earth-ites, in ways that this spoiler does not have room to enumerate. But you asked what would happen if you jumped.

Most importantly, gravity would no longer pull you toward the previous point of the earth's center. You'd be attracted to the center of mass of your hemisphere. In that sense, the "edge" of the earth would be perceived by you as a vast mountainous ridge. Jumping "off the edge" would be like jumping over the ridge of the roof of a house. You might go only about 4' or so, unless you're an Olympic athlete, and then you might continue to slide for a while, or you might be able to stop your slide and come quickly to rest. In any event, your flight would not take you down a sheer, 8000' cliff. It would, however, put you on something like a 45^o slope - both before and after your jump.

Actually, it would not be that symmetrical. You'd be jumping from something like a 30^o slope onto something like a 60^o slope. That probably means friction would not be great enough for you to immediately stop your slide. The good news is that you would not slide all the way to cinder-land. [At least not all the way to the center. The earth does get rather hot, long before you reach the center. Think magma and volcanoes.] The slope would appear to level out as you went along - the pull would always be toward the hemisphere's center. Make a rough sketch to see how that would work. At the [previous] center, the ground would be perceived as level, and apart from incineration you'd be able to stand there on level ground.

Considering that last state, your weight would have changed during your jump/slide. Initially, your weight would have been somewhat less than half your weight now, because the earth's mass was halved during the split and your distance from the center of mass was only slightly increased. As you slid, tho, you'd be getting closer to the center of mass of the hemisphere and gravitational force would increase, at least initially. Hmmm. Or maybe not. How that force changes as you slide would be an interesting calculation. [And discloses a related paradox - which I may write up as a related puzzle ... !]

Hope that gives you some idea. Interesting childhood thought!

[Guest]

Thats a prety damn quastion.But let me tell you what i think.If the earth was ripped in half then something realy huge must have happened.I don't think that

the earth would still keep on turning but the two half would bang with each other and crash eventualy seperating or getting back in one piece.But with all this

happening where would the magma go?A realistic answer is not possible for me.But if we suppose that we have a controled situation of two semispheres

one next to another i can suppose that: (I am not really good at this but someone else might find it)

Calculate the speed somone will reach when he reach the earth centere (About 6,360 km radious) with m*g*h

Then calculate how far will this person go, considering the negative force applied on him by earth.

If the person with the speed he aquired won't be able to get out of earths atmoshpere he will fall back and he will travel less distance each time

eventually being stabilized in earths centere and (propably) stay on air on the total center of the earth

[Guest]

OK, first off, the guy would more than likely be crushed by increasing gravity towards the center. And probably pass out before that. But, for shitz and gigglez, let's just say that the guy falling is invincible. He is superman without the flying and kryptonite weakness. And for all these sick guys who have to get into the incineration and molten lava-_-, lets say it's a hole through a spherical object the size of the earth that has no molten lava. Depending on the aerodynamics of the person falling through the hole, they may reach a speed where, after "bouncing" from one end of the hole to another, they end up in the very center of the hole in a state of suspension. Theoretically, if they happen to have perfect surfaces that would cause no resistance, they may be able to reach a velocity and speed that could "slingshot" them through the other side and into the vast reaches of space. If this where the case, they would continue until they hit something or are smashed into by a passing space rock of some sort.

[bonanova]

Theoretically, ... they may be able to reach a velocity and speed that could "slingshot" them through the other side and into the vast reaches of space. If this where the case, they would continue until they hit something or are smashed into by a passing space rock of some sort.

If that were true, the space program could be run without resorting to rockets.

What a fuel savings!

It requires energy to lift an object at rest above the earth's surface.

[k-man]

If that were true, the space program could be run without resorting to rockets.

What a fuel savings!

It requires energy to lift an object at rest above the earth's surface.

That's correct, but if we had the ability to "orbit" inside the earth around it's center of gravity then we could use the slingshot effect to send rockets into space with a lot less fuel than we need now.

[Guest]

That's correct, but if we had the ability to "orbit" inside the earth around it's center of gravity then we could use the slingshot effect to send rockets into space with a lot less fuel than we need now.

You can't get a gravity-assist from an object you are already either orbiting, or on the surface of. You have to intersect with an object from a different orbit to gain or lose energy from it.

There is a reason, however, why launch sites are always as close to the equator as politics and geography will allow, and why the first maneuver is always to head east(ish)...

[k-man]

You can't get a gravity-assist from an object you are already either orbiting, or on the surface of.

Correct, because you generally can't get "inside" the object. However, consider a rocket on the surface of the earth that can begin its acceleration toward the center of the earth using earth's gravity. It's not much different than a rocket approaching a gravity field of a planet from the outer space.

[Guest]

Correct, because you generally can't get "inside" the object. However, consider a rocket on the surface of the earth that can begin its acceleration toward the center of the earth using earth's gravity. It's not much different than a rocket approaching a gravity field of a planet from the outer space.

No, it still doesn't work. You're only converting potential energy into kinetic energy. You're not gaining anything. The simplest example of a gravity-assist I can think of is analogous to a Hohmann Transfer:

Imagine that you're in a low-Earth-orbit (LEO), and want to instead be in the same orbit as the Moon. You could use a Hohmann transfer. The first burn (perigee kick) puts you into an elliptical orbit that is tangent to both the initial (LEO) and final (Moon) orbits. The second burn (apogee kick) gives you the remaining energy required. However, you could do away with the apogee kick and simply collide with the Moon.

Other, more complex maneuvers are possible (e.g. the Mariner and Voyager missions), but that's the general idea.

Note that the free energy comes from the Moon, not Earth.

(Edit: links)

Edited November 12, 2010 by d3k3

[k-man]

No, it still doesn't work. You're only converting potential energy into kinetic energy. You're not gaining anything. The simplest example of a gravity-assist I can think of is analogous to a Hohmann Transfer:

Imagine that you're in a low-Earth-orbit (LEO), and want to instead be in the same orbit as the Moon. You could use a Hohmann transfer. The first burn (perigee kick) puts you into an elliptical orbit that is tangent to both the initial (LEO) and final (Moon) orbits. The second burn (apogee kick) gives you the remaining energy required. However, you could do away with the apogee kick and simply collide with the Moon.

Other, more complex maneuvers are possible (e.g. the Mariner and Voyager missions), but that's the general idea.

Note that the free energy comes from the Moon, not Earth.

(Edit: links)

Even if the pure slingshot without using the engines is debatable in this scenario you could definitely use the Oberth effect to dramatically reduce the amount of fuel needed to launch rockets into space. The rocket could accelerate from the surface toward the center of the earth without burning any fuel at all and then fire its engines when it reached the center. It could then achieve the escape velocity using a lot less fuel than would require in a conventional launch.

[EventHorizon]

OK, first off, the guy would more than likely be crushed by increasing gravity towards the center. ...

The air pressure would increase as you get closer to the center, but gravity would decrease. I think you are probably thinking of Newton's law of universal gravitation, but that assumes point masses. For instance, if you are at the center of gravity of the earth, then there would be no force of gravity since there would be matter pulling almost equally in all directions... canceling the forces out. Using the equation directly, however, you'd instead expect infinite gravity... but I don't think we've collapsed into a black hole yet (though I could be wrong...).

[Guest]

Even if the pure slingshot without using the engines is debatable in this scenario you could definitely use the Oberth effect to dramatically reduce the amount of fuel needed to launch rockets into space. The rocket could accelerate from the surface toward the center of the earth without burning any fuel at all and then fire its engines when it reached the center. It could then achieve the escape velocity using a lot less fuel than would require in a conventional launch.

It is not debatable. You need to be in a different orbit than the body you get the slingshot from. Period.

You make a good point about the Oberth effect, but I'm not sure how it would work in this situation. One problem is that the math no longer behaves when you're "orbiting" inside the body. For starters, you no longer have a constant gravitational parameter (effectively mu = 0 at r = 0), so you can forget about using vis viva.

[Guest]

Well, it seems to me that one can only fall halfway through the Earth, because then you're falling "out" (or "up"). "Up", "Down" and other directionality is relative in space.

[Akriti]

Nowhere because in space we will fload.