Velocity addition formula in X and Y axes (Relativity)

In summary, the problem at hand is about visualizing a scenario where the light source is at rest in S' but is approaching S. There seems to be confusion about the direction of motion of S and the light source, and whether the direction of v can be changed in the diagram. In order to get the correct answer, the relative positions of the origins of S and S' must also be changed. The problem states v as a speed, not a velocity, so the direction of motion must be determined. There are two possible solutions, one where the cone angle is larger in the opposite direction, but the light is on the outside of the cone and the interior is dark.
  • #1
phantomvommand
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Homework Statement
Please see the attached photos
Relevant Equations
Velocity addition formula (in X and Y axes)
The problem:
Screenshot 2021-03-16 at 11.30.40 AM.png

Visualising the problem (My question is with regards to this):
Screenshot 2021-03-16 at 11.31.26 AM.png

Why is the above set-up correct? In the above diagram, S would be moving at velocity -v relative to S', instead of v. Is this because the question says "speed v", and so we can set the direction as such? Why would the opposing direction of v be incorrect?
 
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  • #2
The rest frame of light source is S'. I think the better way to illustrate would be :

This is an illustration in S. The light cone angle be not 45 degree as drawn in Fig.58 but 30 degree.

or

This is an illustration in S'. The arrow of v on S' axis drawn in Fig.58 be deleted but arrow of opposite direction v should be drawn on S axis.
 
  • #3
mitochan said:
The rest frame of light source is S'. I think the better way to illustrate would be :

This is an illustration in S. The light cone angle be not 45 degree as drawn in Fig.58 but 30 degree.

or

This is an illustration in S'. The allow of v on S' axis drawn in Fig.58 be deleted but allow of opposite direction v should be drawn on S axis.
Yes, The rest frame of light is S'. But S is moving at speed v with respect to S'. In the diagram, it shows S moving at velocity -v with respect to S', why is it not correct to show S moving at velocity v with respect to S'? (which is the same as S' moving at -v with respect to S)
 
  • #4
The diagram shows light source at rest in S' is approaching to S. it will come to origin of S and then surpass it and leave to infinite far away. In all the procedures light cone angle keeps 30 degree in S. So I think you do not have to keen on v or -v, approaching or leaving.
 
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  • #5
mitochan said:
The diagram shows S' is approaching to S. S' will superpose on S and then surpass it and leave to infinite far away.
Why not S' move away from S (move in the other direction)? Where in the question says this cannot be the case?
 
  • #6
Why don't you take v or -v as you like and solve the problem and then change the signature and solve it again?
 
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  • #7
mitochan said:
Why don't you take v or -v as you like and solve the problem and then change the signature and solve it again?
I interpreted it as S' moving to the left, but the light moves to the right. Hence, I do not get the correct answer. So I only swapped the direction of v, but not the direction of the light. Why is this interpretation wrong?
 
  • #8
What is the wrong answer you got ?
 
  • #9
phantomvommand said:
Why is the above set-up correct? In the above diagram, S would be moving at velocity -v relative to S', instead of v. Is this because the question says "speed v", and so we can set the direction as such? Why would the opposing direction of v be incorrect?
Yes, it's because it says "speed" not "velocity".

phantomvommand said:
Yes, The rest frame of light is S'. But S is moving at speed v with respect to S'. In the diagram, it shows S moving at velocity -v with respect to S', why is it not correct to show S moving at velocity v with respect to S'? (which is the same as S' moving at -v with respect to S)
You could, but you'd also have to change the relative positions of the origins of S and S' so that the observer at rest in S approaches the light source.

phantomvommand said:
I interpreted it as S' moving to the left, but the light moves to the right. Hence, I do not get the correct answer. So I only swapped the direction of v, but not the direction of the light. Why is this interpretation wrong?
It's not clear what you did here. So the light is moving in both S and S'?
 
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  • #10
Yes, the problem expresses v as a speed, not a velocity. It is part of the problem to determine the direction of motion of the source in S since the angle gets larger * in the other direction.

* There are actually two solutions to the problem, but with the other solution, the angle of the cone is the same, but the light is on the outside and the interior of the cone is dark.
 
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Related to Velocity addition formula in X and Y axes (Relativity)

1. What is the velocity addition formula in X and Y axes in relativity?

The velocity addition formula in X and Y axes in relativity is a formula used to calculate the combined velocity of an object moving in two different directions, taking into account the effects of special relativity. It is also known as the relativistic velocity addition formula.

2. How is the velocity addition formula in X and Y axes derived?

The velocity addition formula in X and Y axes is derived from the principles of special relativity, specifically the Lorentz transformation equations. These equations describe how time, length, and velocity change for an object moving at high speeds.

3. What is the difference between the velocity addition formula in X and Y axes and the classical velocity addition formula?

The classical velocity addition formula, also known as the Galilean velocity addition formula, only takes into account the velocities of objects in one direction. The velocity addition formula in X and Y axes, on the other hand, takes into account the velocities in two different directions and the effects of special relativity.

4. How does the velocity addition formula in X and Y axes affect our understanding of velocity?

The velocity addition formula in X and Y axes is an important concept in relativity as it helps us understand how velocity is relative to the observer. It also shows that the laws of physics are the same for all observers, regardless of their relative velocities.

5. Can the velocity addition formula in X and Y axes be applied to all types of velocities?

Yes, the velocity addition formula in X and Y axes can be applied to all types of velocities, including velocities close to the speed of light. It is a fundamental concept in relativity and is used in various fields such as physics, astronomy, and engineering.

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