Relativistic Dynamics Problem - Reference Frames

In summary, the primed coordinate system is still moving with the same rightward speed, but 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.
  • #1
Lost1ne
47
1

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?
 

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  • #2
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.
 
  • #3
TSny said:
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.
 
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  • #4
Lost1ne said:
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.
 
  • #5
TSny said:
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.
 

1. What is relativistic dynamics?

Relativistic dynamics is a branch of physics that studies the motion of objects at high speeds, close to the speed of light. It takes into account the effects of special relativity, which describes how time and space are perceived differently for observers moving at different speeds.

2. What is a reference frame in relativistic dynamics?

A reference frame is a set of coordinates used to describe the motion of an object. In relativistic dynamics, it is important to specify the reference frame from which the measurements are being made, as different frames can have different observations of the same event.

3. How is time affected in different reference frames in relativistic dynamics?

According to special relativity, time is relative and can pass at different rates for observers in different frames of reference. This is known as time dilation, and it occurs when objects are moving at high speeds or in strong gravitational fields.

4. Can momentum be conserved in relativistic dynamics?

Yes, momentum is still conserved in relativistic dynamics. However, the equations for calculating momentum and energy in special relativity are different from classical mechanics. The total momentum and energy of a system will be the same in all reference frames, but their individual values may differ.

5. How does the concept of length contraction apply in relativistic dynamics?

Length contraction is the phenomenon where objects appear to be shorter when viewed from a reference frame in motion relative to them. This is a consequence of special relativity and occurs at high speeds. Objects in motion are perceived to be shorter in the direction of their motion by observers in a different reference frame.

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