Relativistic Dynamics Problem - Reference Frames

[Lost1ne]

Homework Statement

Two images are attached. The first image details the problem. The second image has an x',y' coordinate system depiction of the problem.

Homework Equations

The total energy of a particle is defined as E = mc^2, with m = γ*m_0.

The Attempt at a Solution

If the x', y' coordinate system is "linked" to one of the particles (it seems like the left one), why is it that after the inelastic collision takes place we have a larger mass (okay) with a non-zero speed V leftwards (what?)? If our x',y' frame is linked to the left particle, it should always depict the position of that left particle it is attached to as at the origin of the coordinate system and with a velocity of zero, right? (Relative to the left particle, aka from this reference frame, it's the other mass that's doing all the moving, moving with a speed U towards the left particle.) So why is it that after the collision takes place this same reference frame claims that the new larger mass has a non-zero velocity? If the frame is still linked with the original, left mass which is now linked with the other mass, why would the previous result not stand that the position and velocity of the new mass from this same reference frame is zero?

 

[TSny]

The primed frame is "linked" to the left particle only before the collision takes place. The primed frame is an inertial frame, so it cannot slow down relative to the unprimed inertial frame. So, as the collision starts to take place, the particle on the left slows down relative to the unprimed frame while the primed frame keeps going at its original speed relative to the unprimed frame. Thus, once the collision starts, the primed frame is no longer "linked" to the particle on the left.

 

[Lost1ne]

The primed frame is "linked" to the left particle only before the collision takes place. The primed frame is an inertial frame, so it cannot slow down relative to the unprimed inertial frame. So, as the collision starts to take place, the particle on the left slows down relative to the unprimed frame while the primed frame keeps going at its original speed relative to the unprimed frame. Thus, once the collision starts, the primed frame is no longer "linked" to the particle on the left.

That makes sense. So, the "primed" coordinate system is, from the perspective of our "un-primed", x,y frame, is moving with at a rightward speed V and continues to do so to serve as the rest frame of the left particle. However, as it is an inertial frame, it may not accelerate relative to our x,y frame (an additional inertial reference frame) and thus the x', y' frame is constrained to continue to move at that constant, rightwards velocity with speed V. As a result of this, the larger mass after the collision appears to move with a *leftward velocity with speed V from this primed frame while it is viewed at rest from the unprimed frame.

In a sense, our primed shouldn't be thought of as "linked" to the left particle but as simply exhibiting the same velocity as that particle we are trying to construct a rest frame with BEFORE any external force may accelerate that particle. The "linking" notion fails in a situation such as this where our left particle collides with another particle and experiences an acceleration.

 

[TSny]

That makes sense. So, the "primed" coordinate system is, from the perspective of our "un-primed", x,y frame, is moving with at a rightward speed V and continues to do so to serve as the rest frame of the left particle. However, as it is an inertial frame, it may not accelerate relative to our x,y frame (an additional inertial reference frame) and thus the x', y' frame is constrained to continue to move at that constant, rightwards velocity with speed V. As a result of this, the larger mass after the collision appears to move with a rightward velocity with speed V from this primed frame while it is viewed at rest from the unprimed frame.

In a sense, our primed shouldn't be thought of as "linked" to the left particle but as simply exhibiting the same velocity as that particle we are trying to construct a rest frame with BEFORE any external force may accelerate that particle. The "linking" notion fails in a situation such as this where our left particle collides with another particle and experiences an acceleration.

Yes, that all sounds good. But I think you meant to say that in the primed frame, the masses end up moving leftward, not rightward.

 

[Lost1ne]

Yes, that all sounds good. But, I think you meant to say that in the primed frame, the masses end up moving leftward, not rightward.

Yes. Thank you.