Two bodies with rope moving apart

[nitrousuk]

Given two equal mass bodies, moving in opposite directions with equal momentum, attached by rope. When the rope becomes tort, what happens?
Does the momentum cancel out and they become close to at rest?
Or do they move inwards at some rate?

This question came up after watching Gravity :).
This would help settle a debate.
(Please state basic credentials to add weight to arguments)

 

[mikeph]

It depends on the elastic properties of the rope. Total momentum is always conserved, so the two bodies will have the same speed but opposite direction of travel at any moment. If the rope is perfectly elastic they will end up moving towards one another with the same as their original speed (i.e. kinetic energy is conserved). If the rope has losses (like all real ropes) then they will move towards one another with less than their original speed or, in a limiting case, they will come to rest if the rope absorbs all their kinetic energy. But that's unlikely.

 

[nitrousuk]

So assuming there are no elastic properties (perhaps a chain), then they would come to close to rest?

 

[mikeph]

Not exactly. If the chain is not allowed to deform then it cannot even absorb any of the kinetic energy of the bodies (let alone dissipate it), so they will instantly bounce back towards one another as if they both hit concrete walls.

To slow or stop the bodies, the chain must stretch (converting kinetic energy to elastic potential energy), but stretch in a way such that some of this energy cannot be converted back to kinetic energy. For example, internal heating (like a squash ball getting hot), acoustic waves (like when you twang an elastic band). Plastic deformation is an example of a mechanism by which the kinetic energy can be converted into some other less useful form.

 

[jbriggs444]

Not exactly. If the chain is not allowed to deform then it cannot even absorb any of the kinetic energy of the bodies (let alone dissipate it), [...]

Zero deformation multiplied by infinite force -- our physical theories do not make a definite prediction for the energy absorbed when a hypothetical completely rigid chain comes taut. The result is an indeterminate form.

Our theories (Special Relativity in particular) hold that perfect rigidity is not possible.

 

[nitrousuk]

I think the trouble was understanding how the vectors add up.
Given the two initial vectors, equal in opposite directions (say +1 down the z, and -1 along the z). After the binding rope/chain becoming taut, if both masses have their kinetic energy canceled out, as well as accelerated to the same speed in the opposite direction, surely this requires double the oppsite force? Once to cancel it's initial momentum (and bring it to rest) and then the same force again to restore its original speed (but in the opposite direction). Isn't this a doubling of the kinetic energy?

 

[jbriggs444]

I think the trouble was understanding how the vectors add up.
Given the two initial vectors, equal in opposite directions (say +1 down the z, and -1 along the z). After the binding rope/chain becoming taut, if both masses have their kinetic energy canceled out, as well as accelerated to the same speed in the opposite direction, surely this requires double the oppsite force? Once to cancel it's initial momentum (and bring it to rest) and then the same force again to restore its original speed (but in the opposite direction). Isn't this a doubling of the kinetic energy?

In the case you are describing, each object hanging on the end of the rope would experience an impulse twice as large as its momentum. But neither impulse nor force are the same thing as work. It takes work to change kinetic energy and the work done in this case is zero.

 

[dauto]

Would anybody please tell me what happens in the movie so that I can understand this forum?

 

[K^2]

Would anybody please tell me what happens in the movie so that I can understand this forum?

There is a portion of the movie where two astronauts operate while tethered together. One of them has an EVA maneuvering pack, so he's trying to get both of them to the station. Naturally, the whole thing proves rather challenging.

To answer OP, in situation shown in the movie, what you'll have is some recoil. That is, bodies will reach the maximum separation, tether becomes taut, and the two bodies will reverse direction, but moving at a slower speed towards each other than they were moving apart.

Also, keep in mind that in the movie the situation is more complicated due to the fact that tether is not attached at the center of mass of either astronaut. (That'd be hard to do.) So there is also torque on each body, which sends them spinning each time the tether becomes taut.

 

[mikeph]

I think the trouble was understanding how the vectors add up.
Given the two initial vectors, equal in opposite directions (say +1 down the z, and -1 along the z). After the binding rope/chain becoming taut, if both masses have their kinetic energy canceled out, as well as accelerated to the same speed in the opposite direction, surely this requires double the oppsite force? Once to cancel it's initial momentum (and bring it to rest) and then the same force again to restore its original speed (but in the opposite direction). Isn't this a doubling of the kinetic energy?

If you reverse the direction of something with the same speed, the kinetic energy is the same. Whether the cable is able to do that is down to its elastic properties.

Zero deformation multiplied by infinite force -- our physical theories do not make a definite prediction for the energy absorbed when a hypothetical completely rigid chain comes taut. The result is an indeterminate form.

Our theories (Special Relativity in particular) hold that perfect rigidity is not possible.

I'm sorry to say this, but this is a masterclass in missing the point. We are not concerned with the intricacies of special relativity, the difference between perfect rigidity and *nearly perfect* rigidity is irrelevant in the context of the original question.

The question is essentially one of structural mechanics, bringing Einstein into it serves nobody.

 

[jbriggs444]

I'm sorry to say this, but this is a masterclass in missing the point. We are not concerned with the intricacies of special relativity, the difference between perfect rigidity and *nearly perfect* rigidity is irrelevant in the context of the original question.

The question is essentially one of structural mechanics, bringing Einstein into it serves nobody.

"rigid" and "elastic" are not opposites. Rigidity is how strongly an object resists deformation. Elasticity is how well an object returns to its original shape after a deformation. You appear to have brought up the possibility of a chain that is "not allowed to deform" as if that were a synonym for a chain that is perfectly inelastic. I was trying to point out both that complete rigidity does not imply complete inelasticity and that complete rigidity is impossible anyway.

It is also true that near complete rigidity does not imply near complete inelasticity.

 

[nitrousuk]

There is a portion of the movie where two astronauts operate while tethered together. One of them has an EVA maneuvering pack, so he's trying to get both of them to the station. Naturally, the whole thing proves rather challenging.

To answer OP, in situation shown in the movie, what you'll have is some recoil. That is, bodies will reach the maximum separation, tether becomes taut, and the two bodies will reverse direction, but moving at a slower speed towards each other than they were moving apart.

Also, keep in mind that in the movie the situation is more complicated due to the fact that tether is not attached at the center of mass of either astronaut. (That'd be hard to do.) So there is also torque on each body, which sends them spinning each time the tether becomes taut.

Is it the tether itself which absorbs their initial momentum, allowing the kinetic energy to be transferred into movement in the opposite direction?
When you say they move at a slower speed inwards towards each other, roughly how much so? Significantly or marginally? I think the issue with the film was that they would move inwards at exactly the same speed as their initial, with a tether that presumably would be far more rigid than elastic.

I can't help feeling that the tether itself has a significant part in the energy here..
I take it it *must* have some sort of elastic properties in order to reverse the direction? A totally rigid (and theoretical?) tether would not allow transferring of the kinetic energy into inward movement? Conversely, to have perfect transfer to inward momentum (exactly same speed, but inward), would that require close to perfect elasticity in the tether?

Another example that comes to mind, is if say the two masses had small jets attached. Same scenario, same mass, same speed, opposite directions. To have them change direction, and then move inwards, as they would when tethered, wouldn't this scenario require twice the energy as the initial? Say 2 Newtons thrust in opposite direction, to cancel and then reverse the initial 1 Newton of momentum. Which leads me to believe the initial momentum is being transferred into the tether, allowing the kinetic energy to be redirected into inward movement.

Thanks for the answers!

 

[nitrousuk]

Scratch last post. Think I understand it now. Thanks! :)