I am trying to resolve a question about Relativity Theory, with an answer key at hand. The answer given creates more confusion than it resolves, and I am looking for some insights and answers to three very specific questions.
A bullet moves at speed V'. The bullet has length B in its rest-framework. A camera located at a distance, at angle θ in relation to the bullet's linear trajectory, films the bullet. Parallel to the bullet's trajectory and located behind it, is a ruler which is at rest in relation to the camera. What is the length of the bullet as it appears on the film? (note that it takes time for photons to reach the camera and be recorded).
The answer key gives the following:
The photons indicating the start and the end of the bullet respectively need to both reach the camera simultaneously. This means they will leave the bullet at different times, given that the photon reflected at the end of the bullet has a longer trajectory to reach the camera, that is to say longer by a factor $b'·cosθ$. See illustration:
Here come my three problems:
1. According to the Lorentz transformations that I learnt:
$x'=γ(x-vt)$
Why - instead - do we find $x'=γ(x-βt)$ in this instance?
2. When I isolate $b'$ from the first equation in the illustration, I end up with $b'=\frac{b(1-βcosθ)}{γ}$, rather than $b'=\frac{b}{γ(1-βcosθ)}$ .What am I missing out on?
3. Why does the answer key use $b·cosθ$ and not $b'·cosθ$ in the formulae?
Many thanks!
