A few special relativity questions before my exam

[mewmew]

Ok so my exam is here tomorrow and I don't feel like I have the grasp on SR I would like to have, most of the books I ordered recently got here so I haven't had time to read them yet. So if I could get any help with some questions I would really appreciate it.

1.) When using the general Doppler shift formula $$\nu'=\frac{v_0}{\gamma(1-\beta Cos\theta)}$$ I am a little confused on how its derived by resolving the problem into radial and transverse motion. I would think just the radial would give the above result as the only difference is that the Cos that is taken as the velocity is only a component of the full velocity source velocity. I don't know what to do with the time dilated only transverse part though as it doesn't seem to be needed. Hopefully that makes sense.

2.) When thinking of which to use, full Lorenz transformations or just time dilation/length contraction formulas what is the best way to decide which to use? It is giving me a bit of trouble even though it seems to be pretty straight forward.

Thanks for any help, hopefully I can have everything straight out by tomorow.

 

[Jhero]

I understand your concern about your upcoming exam and your confusion with Special Relativity (SR). Let me try to provide some clarification and assistance with your questions.

1.) The general Doppler shift formula is derived by considering the relative motion between the source of the wave (e.g. a moving object emitting sound or light) and the observer. This relative motion can be broken down into two components: a radial component (along the line of sight between the source and observer) and a transverse component (perpendicular to the line of sight). The radial component is responsible for the frequency shift, while the transverse component is responsible for the time dilation. The full formula takes into account both components, hence the inclusion of the Cosine term. The time dilated transverse part is needed because it affects the observed frequency due to the relative motion between the source and observer.

2.) The choice between using the full Lorentz transformations or just the time dilation/length contraction formulas depends on the specific problem you are trying to solve. The full Lorentz transformations are more general and can be applied to any situation involving relativistic motion. However, if you are only concerned with the effects of time dilation and length contraction, then it is more efficient to use the simplified formulas. It is important to understand the underlying concepts and equations in order to determine which approach is best for a given problem.

I hope this helps clarify your understanding of SR and the Doppler shift formula. Good luck on your exam tomorrow! Remember to take your time and carefully analyze each question. Don't hesitate to ask for help if you need it. Best of luck!