Integration when transform to center of mass frame

Click For Summary

Discussion Overview

The discussion revolves around the evaluation of a specific integral in the context of transforming to the center of mass frame. Participants are exploring the limits of integration for the variables involved, particularly in relation to the center of mass frame, rather than solving the integral itself.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in determining the correct limits for the integral in the center of mass frame after substituting variables.
  • Another participant shares a method of fixing a point and integrating over a shell radius, suggesting it was straightforward but may not directly address the original question.
  • A participant clarifies their interest in the limits of integral variables in the center of mass frame, indicating that the previous response did not align with their query.
  • There is a concern raised about the complexity of determining limits based on the orientation of the line when using a midpoint near shell boundaries.
  • One participant reiterates their focus on understanding the full expression of the limits rather than solving the integral itself.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the correct approach to determining the limits of integration in the center of mass frame, with multiple viewpoints and methods being discussed.

Contextual Notes

There are indications of uncertainty regarding the dependence of the limits on the orientation of the line and the complexity of the problem, which may not have been fully resolved in the discussion.

babylonia
Messages
9
Reaction score
0
Hi,

I am having some difficulty doing the integral
∫d[itex]^{3}[/itex]v1d[itex]^{3}[/itex]v2 | [itex]\overline{v1}[/itex]-[itex]\overline{v2}[/itex]|, where u1[itex]\leq[/itex]|v1|,|v2|[itex]\leq[/itex]u2, and [itex]\overline{v1}[/itex] means vectors.

It seems better to evaluate it in the center of mass frame, by substitution [itex]\overline{v1}[/itex]+[itex]\overline{v2}[/itex]=[itex]\overline{V}[/itex], and [itex]\overline{v1}[/itex]-[itex]\overline{v2}[/itex]=2[itex]\overline{v}[/itex],
However, I'm not sure what are the correct integral limits for |V| and |v|.

Can anybody give me some help? Really appreciate deeply.

Thanks.
 
Physics news on Phys.org
I tried fixing a point at a from the origin, letting another point range over a shell radius r < a, integrating the distance between them. Wasn't too difficult.
 
Hi,

Thanks for your reply, but I'm not sure you are replying to my post? What you mentioned does not seem to be the thing I was asking? I'm more interested to know the limits of integral variables in CM frame instead of working out this particular integral. Could you tell more details even if you think it very easy?

Thanks.
 
Last edited:
babylonia said:
I'm more interested to know the limits of integral variables in CM frame instead of working out this particular integral.
Ah, yes. I thought about that for two seconds and decided it was so nasty it couldn't be the right approach. Imagine picking a midpoint near one of the shell boundaries. The range for the separation then depends in a very awkward way on the orientation of the line.
If I've convinced you of that, try my way and let me know if you need more help.
 
Thanks for you reply.

I have no difficulty working out this particular integral, since I actually picked an easy form just to present my question about those limits. My problem is to know the full expression of those limits.

Thanks any way.


haruspex said:
Ah, yes. I thought about that for two seconds and decided it was so nasty it couldn't be the right approach. Imagine picking a midpoint near one of the shell boundaries. The range for the separation then depends in a very awkward way on the orientation of the line.
If I've convinced you of that, try my way and let me know if you need more help.
 

Similar threads

Undergrad Two-Mass Oscillator: Plotting Amplitudes Over Frequency in Hertz
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Question about the velocity of the center of mass reference frame
  • · Replies 21 ·
Replies
21
Views
2K
Graduate Motorcycle, the force exerted upon from collision
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Velocity transformation between frames rotating relative to one another
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Coefficient of restitution in different frames of references
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Torque about center of mass due to Coriolis force, is it 0?
  • · Replies 2 ·
Replies
2
Views
2K
Kinetic energy in center of mass reference frame
  • · Replies 3 ·
Replies
3
Views
2K
Lorentz transformation & relativistic momentum
  • · Replies 1 ·
Replies
1
Views
2K
What are the Lorentz transformation tensors used for?
  • · Replies 5 ·
Replies
5
Views
2K
What is the correct transformation for a 4-vector in special relativity?
  • · Replies 3 ·
Replies
3
Views
2K