Relative Velocity: Jack & Particle - v to V

[Izzhov]

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?

 

[Nugatory]

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.

 

[Dale]

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.

 

[Izzhov]

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.

 

[WannabeNewton]

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.

 

[Dale]

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.

 

[Izzhov]

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?

 

[WannabeNewton]

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.