I Questions About Acceleration and Jerk in Orbits

[AdrianGriff]

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

 

[Orodruin]

This is not generally true. Are there any additional assumptions you have forgotten to mention? What type of orbit are you referring to?

 

[AdrianGriff]

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

 

[Orodruin]

For a circular orbit it is certainly not true, as you have already concluded.

 

[AdrianGriff]

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.

 

[Orodruin]

Could you invest a bit of time into helping me understand this question?

It seems like a straightforward application of the work-energy theorem.

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.

 

[AdrianGriff]

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 :)