Velocity needed to escape gravitational potential

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Discussion Overview

The discussion revolves around determining the velocity required for an asteroid to escape a planet's gravitational pull. Participants explore the application of conservation of energy and angular momentum in this context, examining the behavior of radial and angular velocities as the distance from the planet approaches infinity.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant describes using conservation of energy and angular momentum to derive an expression for radial velocity, questioning its validity when the initial velocity is not purely radial.
  • Another participant asserts that at infinity, the velocity vector points radially outward.
  • A participant agrees with the initial approach and suggests finding an expression for angular velocity, prompting further inquiry into its behavior at infinity.
  • It is noted that while angular velocity approaches zero as distance increases, this does not necessarily imply that tangential velocity also approaches zero.
  • Concerns are raised about the implications of angular velocity tending to zero and its relation to the direction of travel, questioning the relevance of angular velocity in the initial inquiry.

Areas of Agreement / Disagreement

Participants express differing views on the implications of angular velocity and its relationship to tangential velocity, indicating that multiple competing perspectives remain on the topic.

Contextual Notes

The discussion highlights potential limitations in reasoning about the relationship between angular and tangential velocities, as well as the assumptions regarding the direction of velocity at infinity.

Fibo112
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I am solving a problem where I need to decide if an asteroids velocity is high enough to escape the planets gravitational pull. The way I did it was use conservation of energy and angular momentum to find an expression for the radial velocity and show that it remains positive as r tends to infinity. In the solution all that was done was argue that the sum of potential and kinetic energy is positive. I can see how this would work if v0 is pointing radially outward but I don't see why is holds generally. Does it and if so why.
 
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At infinity, what direction does the [other end] of the velocity vector point?
 
The velocity will point radially outward at infinity
 
Fibo112 said:
The way I did it was use conservation of energy and angular momentum to find an expression for the radial velocity and show that it remains positive as r tends to infinity.
That is a good approach. You can probably use the same approach to find an expression for the angular velocity. What does it tend to?
 
Well the angular velocity will go to zero in any case where r goes to infinity
 
Dale said:
That is a good approach. You can probably use the same approach to find an expression for the angular velocity. What does it tend to?
An angular velocity that tends to zero is not sufficient to guarantee that the tangential velocity also tends to zero. It may be better to reason based on conservation of angular momentum that the tangential velocity must tend to zero.

Further, an angular velocity that tends to zero is not sufficient to guarantee the existence of a limiting direction of travel.

But then I am not completely certain why @Dale is asking you to consider angular velocity in the first place.
 

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