Relativistic travel: Looking ahead vs back If I stand at the middle of a bus traveling at 0.5c, will the front half of the bus appear to me shorter than the back? and, does it make any difference if there's air inside the cabin versus a vacuum?
Please note: simply attaching a reference frame to the bus and call it a day does not cut it! There is a certain distance to the front and back walls, and what the observer sees is a "delayed" version of those objects, based on the time it takes light to travel that distance. During that time, the observer changes position. So the "delayed" version of the front wall should appear closer to the "current" position of the observer. Similarly, the back wall should appear recessed.
 A: If you are moving with the bus, the bus will appear to you to be at rest- you will not see any relativistic effects whatsoever inside the bus.
It is easy to go wrong when approaching relativity for the first time. Once you do understand it, it all makes sense, but it conflicts with out common sense notions of space and time, and you have be prepared to abandon those if you are to make headway with the subject.
Your question implicitly assumes that there is such a thing as an absolute point in space, or, equivalently, that there is an absolute frame of reference, neither of which are true.
Specifically, let's suppose that as the front of your bus passes me as I stand on the road, the driver flashes a light towards you at the centre of the bus. You seem to think that in that case, the source of the light was the point next to me on the road where the driver was when they flashed the light, and that the driver has since moved on beyond that point and so is no longer at the origin of the light flash. That's not quite true. In SR every reference frame is equally valid for the purpose of calculating the speed of light. In my reference frame, the spatial origin of the flash was indeed the point on the road where the driver was at the time of the flash, but that's not true in your frame of reference. In yours, the spatial origin of the flash was the driver, who remains a fixed distance from you. So, while you, in the moving bus, are moving towards what I consider the origin of the flash, you are stationary relative to the origin of the flash in your frame.
I will give you another example to drive home the point. Suppose you and I are standing together next to a bulb that flashes light which heads off in all directions. If you walk after the light to the right, while I walk after the light to the left, we are each entitled to consider ourselves to be still at the centre of the expanding sphere of light. That might sound entirely contrary to the common sense view that the centre of the sphere must be the place where we were standing when the bulb flashed, but that common sense idea turns out to be just an approximation that is almost exact where low relative speeds are involved but breaks down completely when very high speeds are involved.
I hope you continue to work on SR, as it is conceptually very compelling once you get the hang of it, but if you are like most people (myself included) you will find it very hard to grasp if you keep trying to reconcile it with common sense presuppositions about space and time.
A: The very basic fact is that motion is relative. It is meaningless to say that the bus "travels at 0.5c"; you have to specify – with respect to what? (The Earth? The Sun? The centre of the galaxy? Something else?)
The problem, as currently stated, consists of only two objects – the bus, and the person in the middle of the bus. They are motionless with respect to each other. It is also not true to say that "the observer changes position" – the observer does not change position with respect to the only other object present in the set-up.
Whether or not the whole combination (bus+observer) happens to move with some constant velocity with respect to some third (unspecified) body, does not play a role at all. If the bus happens to move with 0.5c with respect to the Earth, say, we may as well say that the bus is stationary, and the Earth moves with 0.5c (in the opposite direction).
To conclude, the observer will not see a difference looking forward or back, as the observer is not in motion with respect to the bus.
A: Attaching a reference from to the bus and calling it a day does indeed cut it.
The bus is at rest, you're in the middle at the origin. If the bus is length $L$, you see the front and back as they were $L/(2c)$ in the past. If the bus has air in it, then the delay is $L/(2c) \times (1 + \eta)$ where:
$$ \eta = 273 \times 10^{-6} $$
Note that the claimed speed of the bus, $c/2$, is not part of the problem (because there is no absolute reference frame).
