Questions About Acceleration and Jerk in Orbits

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Discussion Overview

The discussion revolves around the relationship between acceleration and jerk in the context of orbital mechanics, specifically questioning the validity of the equation ##\vec a \cdot \vec a' = |a||a'|##. Participants explore the implications of this equation in circular orbits and the conditions under which it may or may not hold true.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Adrian questions the validity of the equation ##\vec a \cdot \vec a' = |a||a'|##, suggesting that the angle between the vectors leads to a contradiction in circular orbits.
  • Some participants assert that the equation is not generally true and inquire about additional assumptions regarding the type of orbit being discussed.
  • One participant mentions that the equation may only hold under specific conditions, such as a perfectly circular orbit and no change in mass of the satellite.
  • Several participants agree that for a circular orbit, the initial conclusion drawn by Adrian is correct, indicating that the equation does not hold.
  • Adrian expresses a desire for further clarification and assistance in understanding the topic, indicating a personal struggle with the concept.
  • Another participant suggests that the question may relate to the work-energy theorem, although this is not universally accepted as a resolution to Adrian's concerns.

Areas of Agreement / Disagreement

Participants generally agree that the equation does not hold true for circular orbits, but there is no consensus on the broader applicability of the equation or the assumptions required for its validity.

Contextual Notes

Participants note the importance of defining the type of orbit and the conditions under which the equation is considered, highlighting potential limitations in the discussion.

AdrianGriff
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In orbits it is said that ##\vec a \cdot \vec a' = |a||a'|##

How is this possible? Two vectors multiply to get scalars, and yet we cannot do the dot product literally because we do not know either of the components of ##\vec a## or ##\vec a'##.

Nor does the Angle Between Vectors Formula work because
$$ \vec a \cdot \vec a' = |a||a'| cos (\theta)$$

And if the acceleration vector is towards the center of the orbit, and jerk, or ##a'## is orthogonal to ##a## and tangent to the circle, opposite of ##v##, then ##\theta = \pi/2##. And if that is the case, then ##cos (\pi/2) = 0##, and as such, ##\vec a \cdot \vec a' ≠ |a||a'|## but rather ##\vec a \cdot \vec a' = 0##

So, how is it possible that ##\vec a \cdot \vec a' = |a||a'|##?

Thank you for your help!
- Adrian
 
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This is not generally true. Are there any additional assumptions you have forgotten to mention? What type of orbit are you referring to?
 
Orodruin said:
This is not generally true. Are there any additional assumptions you have forgotten to mention? What type of orbit are you referring to?
Well this is only a small step in deriving the conservation of mechanical/orbital energy, ##ξ## provided in The Fundamentals of Astrodynamics by Roger R. Bate, But the only extra information that I can think that would be important is that:
1) The orbit is perfectly circular
2) There is no ##Δmass## of the satellite orbiting
 
For a circular orbit it is certainly not true, as you have already concluded.
 
Orodruin said:
For a circular orbit it is certainly not true, as you have already concluded.
Could you invest a bit of time into helping me understand this question? I don't mean to ask too much, but this is a pestering and burning question. Perhaps it is too out of my grasp (I am only 18, still in high school), but regardless, I would like to know why it is the way it is. I could send you the link to an online PDF version of the book so you could see what is going on in more detail?

If not that is fine, I've just been hung up on this idea for days.
 
AdrianGriff said:
Could you invest a bit of time into helping me understand this question?
It seems like a straightforward application of the work-energy theorem.

AdrianGriff said:
I could send you the link to an online PDF version of the book so you could see what is going on in more detail?

If not that is fine, I've just been hung up on this idea for days.
This would not be fine. Copyright violation is against PF rules.
 
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Orodruin said:
It seems like a straightforward application of the work-energy theorem.This would not be fine. Copyright violation is against PF rules.

Oh, okay. Thank you :)
 

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