Relative Velocity: Jack & Particle - v to V

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Discussion Overview

The discussion revolves around the concept of relative velocity, particularly in the context of different reference frames. Participants explore how an observer's speed is perceived from another moving observer's frame, specifically addressing scenarios involving constant speed and circular motion. The scope includes theoretical considerations and the implications of non-inertial frames.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks how fast Jack would observe them moving if Jack is jogging past at speed v, questioning if the answer is v and which Lorentz transformation applies if it is not.
  • Another participant asserts that the answer is v, suggesting that the same speed is observed in both frames.
  • A different participant challenges this by stating that the question about the particle's rest frame is incomplete, emphasizing the lack of a standard meaning for "the rest frame of the particle" in non-inertial contexts.
  • One participant proposes defining the particle's frame as one that coincides with the particle at all times, arguing that this would allow for measuring the observer's circular motion around the particle.
  • Another participant introduces the concept of momentarily co-moving reference frames, explaining that these frames differ at each point along the particle's circular path.
  • There is a reiteration that the particle will measure the observer moving circularly around it, but the specifics of the transformation need to be clarified to avoid ambiguity regarding length scale and simultaneity conventions.
  • One participant concludes that from each momentarily co-moving frame, the particle will observe the observer moving with a velocity opposite to that of the co-moving frame, maintaining the same speed in circular motion from the particle's perspective.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the rest frame for a particle in circular motion, with some asserting that the same speed is observed while others highlight the complexities introduced by non-inertial frames. The discussion remains unresolved regarding the implications of these differing interpretations.

Contextual Notes

Participants note that the definitions of frames and transformations are crucial for clarity, particularly in non-inertial contexts, and that assumptions about simultaneity and scale are not explicitly stated, leading to potential ambiguities.

Izzhov
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First, a really basic one: if Jack is jogging past me with a constant speed, and I observe that speed to be v in my frame, with what speed will Jack observe me to be moving, as measured from his own frame? If the answer is not v, which Lorentz transformation do I use to derive it?

Second, a slightly more advanced one: say I'm in a lab and I measure a particle to be moving in a circle, in which I am in the direct center, with constant speed v. In the rest frame of the particle, what speed would it measure me to be moving around it? If this answer differs from the first question, whence the difference?
 
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Izzhov said:
First, a really basic one: if Jack is jogging past me with a constant speed, and I observe that speed to be v in my frame, with what speed will Jack observe me to be moving, as measured from his own frame? If the answer is not v, which Lorentz transformation do I use to derive it?
It's v.

Second, a slightly more advanced one: say I'm in a lab and I measure a particle to be moving in a circle, in which I am in the direct center, with constant speed v. In the rest frame of the particle, what speed would it measure me to be moving around it? If this answer differs from the first question, whence the difference?
Same answer.
 
Nugatory said:
Izzhov said:
Second, a slightly more advanced one: say I'm in a lab and I measure a particle to be moving in a circle, in which I am in the direct center, with constant speed v. In the rest frame of the particle, what speed would it measure me to be moving around it? If this answer differs from the first question, whence the difference?
Same answer.
Actually, I wouldn't give the same answer. The problem is that the question is not complete. There is no standard meaning to "the rest frame of the particle" when the particle is non-inertial. You would have to define that frame before you could answer the question.
 
DaleSpam said:
Actually, I wouldn't give the same answer. The problem is that the question is not complete. There is no standard meaning to "the rest frame of the particle" when the particle is non-inertial. You would have to define that frame before you could answer the question.

In this context, I would define it as a frame whose origin coincides with the particle at all times and whose axes are parallel to the lab frame's axes. Thus the particle would measure me moving circularly around it.
 
There is no single "rest frame" for a particle moving in circular motion about a central observer. What you can do is, at each point on the particle's trajectory, attach an inertial frame to the particle whose velocity coincides with that of the particle, at that point. Such a frame is called a momentarily co-moving reference frame. For circular motion, the momentarily co-moving reference frames will necessarily be different at each point on the particle's trajectory.
 
Izzhov said:
In this context, I would define it as a frame whose origin coincides with the particle at all times and whose axes are parallel to the lab frame's axes. Thus the particle would measure me moving circularly around it.
The only real way to specify this is to write down the transform from the inertial frame. Otherwise a specification like the one you mentioned leaves the length scale, time scale, and simultaneity convention ambiguous.
 
WannabeNewton said:
There is no single "rest frame" for a particle moving in circular motion about a central observer. What you can do is, at each point on the particle's trajectory, attach an inertial frame to the particle whose velocity coincides with that of the particle, at that point. Such a frame is called a momentarily co-moving reference frame. For circular motion, the momentarily co-moving reference frames will necessarily be different at each point on the particle's trajectory.

Ok, so from each of these co-moving frames, with what velocity will the particle measure me to be traveling circularly around it?
 
You already answered that. At each instant, in a momentarily co-moving inertial frame, the particle will see you move with a velocity opposite from that of the co-moving inertial frame. So if we keep moving along with the particle, you will just be moving in a circle with the same speed from the particle's perspective.
 

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