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Thread: Terminal Velocity without Drag?

  1. #1 Terminal Velocity without Drag? 
    Forum Freshman ClaimingLight's Avatar
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    Is there a 'terminal velocity' for the gravity of an object in an ideal environment free from drag? That is to say: if you were falling downward in a vacuum tube with a limitless length, would you perpetually accelerate or would you eventually be moving with a velocity equal to the pulling force?

    I'm of the mind that Earth can only pull you down so hard. After a while, you won't accelerate any further because you're moving as fast as Earth can push you.


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    Brassica oleracea Strange's Avatar
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    You will never stop accelerating but there is a maximum velocity. This is due to the fact that you can only accelerate for so long before you hit the ground. If you were to fall to earth from an infinite distance (ignoring all other gravitational objects) you would hit the ground at escape velocity (I think).

    If you could fall for ever in the same gravitational field then you would approach the speed of light.


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    Forum Professor jrmonroe's Avatar
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    Quote Originally Posted by Strange View Post
    the speed of light.
    Rats, I wanted to say c.

    On the other hand, what if an object began at rest at an infinite distance (say, the edge of universe), and only Earth and its gravitation existed. What velocity would it reach if also allowed to travel down a tube to the center of the Earth. Perhaps having to ignore c, there would be a limit to the velocity when it reaches the center of the Earth.
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    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by jrmonroe View Post
    On the other hand, what if an object began at rest at an infinite distance (say, the edge of universe), and only Earth and its gravitation existed. What velocity would it reach if also allowed to travel down a tube to the center of the Earth. Perhaps having to ignore c, there would be a limit to the velocity when it reaches the center of the Earth.
    Interesting question. Once it passed the surface of the Earth the force (and therefore acceleration) would decrease. But it would still be accelerating. Until it reached the center when the acceleration would be zero, and then it would start decelerating (reaching zero velocity at infinity).

    I don't know what the velocity would be at the center. That would require integration of the force, I assume. I'll pass
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  6. #5  
    Administrator KALSTER's Avatar
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    I think it is as Strange says, you'll reach escape velocity. Escape velocity is defined as the initial speed needed to have you decelerate so that you would reach zero speed at an infinite distance.

    Edit: Oh yeah, you'll accelerate further until the centre and then tart decelerating again until yo reach zero at infinity in the opposite direction. I imagine you'd need the Shell theorem, requiring integration as Strange said. Yikes.
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    Thinking of a vacuum tube from the end of the universe to the center of earth, the object will oscillate back and forth about the center of mass of the tube-earth system ( Such a long tube will have a lot of mass.)

    If the mass of the tube isint taken into account, a velocity of almost c must be the terminal velocity.
    Also about claiming light's doubt
    Quote Originally Posted by ClaimingLight View Post
    I'm of the mind that Earth can only pull you down so hard. After a while, you won't accelerate any further because you're moving as fast as Earth can push you.
    Earth shall keep pulling it down with the same force no matter what is the velocity ( Of course , you wont accelerate after reaching c)
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  8. #7  
    Administrator KALSTER's Avatar
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    If the mass of the tube isint taken into account, a velocity of almost c must be the terminal velocity.
    If the mass starts falling from an infinite distance, it would reach the escape velocity from the centre of the earth. That is very far removed from C. That would just be the maximum velocity attainable though, not a terminal velocity.

    Edit: Another thing to think about, is that if the earth and that mass were the only masses in a spherical or other unbounded universe, there would be a Lagrange point. No?
    Last edited by KALSTER; June 19th, 2012 at 06:34 AM. Reason: (un)bounded
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  9. #8  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by KALSTER View Post
    it would reach the escape velocity from the centre of the earth
    Although you can't use the standard equation for escape velocity () as r=0 at the center.

    That would just be the maximum velocity attainable though, not a terminal velocity.
    Good distinction to make.

    Edit: Another thing to think about, is that if the earth and that mass were the only masses in a spherical or other bounded universe, there would be a Lagrange point. No?
    They would also both accelerate towards their common center of mass.
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  10. #9  
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    The center of mass of the tube - Earth structure could be the Lagrange point.
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  11. #10  
    Administrator KALSTER's Avatar
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    @ Strange:

    Ok, but imagine a spherical universe. There should be two points where you could place the two masses where the forces would cancel out.

    Imagine an actual sphere with two charged balls on the surface with zero friction. You could in theory place them at an exact point where opposing forces would cancel out. Lagrange points. What am I missing?
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  12. #11  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by KALSTER View Post
    @ Strange:

    Ok, but imagine a spherical universe. There should be two points where you could place the two masses where the forces would cancel out.

    Imagine an actual sphere with two charged balls on the surface with zero friction. You could in theory place them at an exact point where opposing forces would cancel out. Lagrange points. What am I missing?
    I see what you mean; if the universe was finite (and unbounded) then there would be a maximum distance apart they could be and the forces should balance at that point. I guess it would require someone with a better understanding of 4D space-time manifolds to say for sure.

    At some point the difference between "infinite" and "a long way away" becomes irrelevant for the original question, though.
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  13. #12  
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    So we're saying that gravity has a limitless pulling potential, then? That seems counter-intuitive when compared to other examples of F.

    EDIT: I know that special relativity is a possible answer, but that's also a counter-intuitive notion. Are there any other ways of thinking about it that don't include spacetime?
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  14. #13  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by ClaimingLight View Post
    So we're saying that gravity has a limitless pulling potential, then? That seems counter-intuitive when compared to other examples of F.
    Gravity falls off as distance squared. I'm not sure what you mean by "other forces". As far as I know, only the strong nuclear force has a limited range.

    EDIT: I know that special relativity is a possible answer, but that's also a counter-intuitive notion. Are there any other ways of thinking about it that don't include spacetime?
    Apart from the ultimate limit of c, all of the above can be understood from Newtonian gravity.
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    Universal Mind John Galt's Avatar
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    I've been following this thread, but only just noticed it was in Philosophy, so I've moved it to Natural Philosophy, or Physics as it is now known.
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    ClaimingLight: Ignoring the atmosphere along with other celestial bodies, an object falling to Earth from a very great distance would hit the ground at escape velocity, which is circa 11km/s. Maybe you're thinking of an object falling into a black hole from a very great distance? Note though that there's a little problem with that, wherein the coordinate speed of light varies in a non-inertial reference frame.

    Re other examples of F, you could do something similar for an electron and a proton. There is no place where the field/interaction/attraction stops. It just gets less noticeable with distance and ends up being swamped.

    Kalster/Strange: we have no evidence that the universe is a hypersphere. Evidence from say WMAP suggests that space is flat. We don't have any evidence that's it's infinite either.
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  17. #16  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by Farsight View Post
    We don't have any evidence that's it's infinite either.
    We don't have any evidence that it is finite, either. But let's not get too far off track...
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    > I'm of the mind that Earth can only pull you down so hard.

    But that pull is
    continuous, so the acceleration is continuous.
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  19. #18  
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    Quote Originally Posted by Farsight View Post
    ClaimingLight: Ignoring the atmosphere along with other celestial bodies, an object falling to Earth from a very great distance would hit the ground at escape velocity, which is circa 11km/s. Maybe you're thinking of an object falling into a black hole from a very great distance? Note though that there's a little problem with that, wherein the coordinate speed of light varies in a non-inertial reference frame.

    Re other examples of F, you could do something similar for an electron and a proton. There is no place where the field/interaction/attraction stops. It just gets less noticeable with distance and ends up being swamped.

    Kalster/Strange: we have no evidence that the universe is a hypersphere. Evidence from say WMAP suggests that space is flat. We don't have any evidence that's it's infinite either.
    That makes sense. Say a rocket ship is moving at a sustained force of X away from the gravity of a celestial body, which is pulling at a sustained force of Y. You stay still.

    Take that X and put your rocket ship in a perfect vacuum. The acceleration gained from force X would have diminishing returns over time, right? If it does, then the same should be true for force Y.
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  20. #19  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by ClaimingLight View Post
    That makes sense. Say a rocket ship is moving at a sustained force of X away from the gravity of a celestial body, which is pulling at a sustained force of Y. You stay still.

    Take that X and put your rocket ship in a perfect vacuum. The acceleration gained from force X would have diminishing returns over time, right? If it does, then the same should be true for force Y.
    I don't understand why the "perfect vacuum" is relevant?

    If the force from the rocket equals the force of gravity (i.e. X = Y at the position that the rocket is) then it will just hover.

    If X is greater then it will move away, in other words it will start accelerating due to the difference between X and Y (a = (X-Y)/M). But of course, Y will be decreasing (inverse square law) while X is constant and so the rate of acceleration will also increase.

    At a suitably large distance you can consider Y to be negligible and the acceleration will be pretty much constant (a ~ X/M). Of course if you then turn the rocket off, the now tiny force of Y will start slowing you down. Unless you have exceeded escape velocity, it will eventual slow you down to zero and pull you back to the planet. However far away you are.
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