A couple of weeks ago I had the misfortune to be on a bus which had an accident. I wasn't hurt, because I was safely seated, which is more than I can say for one unfortunate passenger who was still on his way to his seat at the time. It wasn't a high-speed event – I'd guess we were doing about 10 km/h. We had just pulled away from a bus stop, when a car that had been parked a few metres in front of the bus decided to pull out into the road right in front of us. The driver hits the breaks hard, and, as a result, the fellow passenger ends up in a heap on the floor at the front of the bus.
While the cause of the crash I would say rests firmly with the driver of the car that pulled out, that's little comfort to the poor guy with blood dripping from a wound on his head, down the back of his shirt, which is probably now dyed a nice shade of maroon. Standing on buses is pretty dangerous, even at low speed. I do think the driver should have waited till everyone was seated before pulling away.
So, from a physics perspective, what happened? One can explain this in two ways. There's the 'inertial' approach, as explained by the witness on the side of the road: The bus stopped, but the guy standing, who has inertia, carried on. Then there's my viewpoint, from inside the bus. Everything experiences a sudden acceleration forward. This causes the passenger to lose his balance, and down he goes.
This forward acceleration, from the perspective of the person on the bus, is called an apparent force. It arises because the frame-of-reference, the bus, isn't an inertial frame. That is, it's accelerating (or, in this case, decelerating). It's called 'apparent' because the person on the side of the road wouldn't see it in this way; it only becomes apparent if the observer is in the accelerating frame of reference. It might be termed an 'apparent' force, but for the person on the bus it's a very real push forwards, one that splatted him on the floor and would have given the bus cleaners a more interesting job that usual. It's the same kind of thing as centrifugal force (yes, the 'f' word), which one experiences when going round corners. To the person in the object that is doing the moving, the force is a very real thing (ask the racing car driver). But to everyone else, it doesn't actually exist.
Apparent forces are pretty hard to teach (I've just been doing it), but I think the key is really to emphasize that they are there only to the observer who is in the accelerating frame.
What happened to the passenger? Against the advice of everyone around him, including me, he refused to be taken to a medical centre, which was only a few hundred metres from the place of the incident, and insisted on carrying on the journey to his destination. Possibly if he'd been able to see the back of his head he might have thought differently. One shudders to think of the consequences at 50 km/h. Seat belts in buses? Yes please.