Deriving Lorentz Transformations for Moving Reference Frames

[B4cklfip]

Homework Statement

Consider three frames Σ (x, y, z, t), Σ' (x', y', z', t'), and Σ'' (x'', y'', z'', t'') whose x, y, and z axes are parallel at each point in time stay. Σ' moves relative to Σ with velocity v1 along the x-axis. The system Σ'' moves relative to Σ' with the velocity v2 along the x'-axis. Determine the Lorentz transformation and the relative velocity between the reference systems Σ and Σ''. Compare this speed with the value that would follow from the non-relativistic calculation.

Relevant Equations

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Problem Statement: Consider three frames Σ (x, y, z, t), Σ' (x', y', z', t'), and Σ'' (x'', y'', z'', t'') whose x, y, and z axes are parallel at each point in time stay. Σ' moves relative to Σ with velocity v1 along the x-axis. The system Σ'' moves relative to Σ' with the velocity v2 along the x'-axis. Determine the Lorentz transformation and the relative velocity between the reference systems Σ and Σ''. Compare this speed with the value that would follow from the non-relativistic calculation.
Relevant Equations: -

Hello PhysicsForum,

until now i had not had any idea how to solve this problem.
Maybe someone can give me a hint or an approach how to get a solution. :)

 

[TSny]

If you write out the transformation between the primed and double-primed systems, you will have a relation between the primed and double-primed coordinates. Similarly, for the transformation between the unprimed and primed systems. The rest should just be algebra.

 

[B4cklfip]

I have tried to calculate it now...

 

[B4cklfip]

This is my result. Could someone take a look and tell me if it is right please?

 

[TSny]

I think your calculation is OK. But, from the way that I interpret the problem statement, I think that they want you to derive the transformation between Σ and Σ'' starting with the transformation between Σ and Σ' and the transformation between Σ' and Σ''. This will lead to an expression for the velocity of Σ'' relative to Σ in terms of v1 and v2 given in the problem. The algebraic manipulations will be very similar to what you have done.