They assume a perfectly frictionless tunnel, and that gravity is the only force acting on the traveller. That's not the only difficulty in realizing such a scheme, though...the heat and pressure deep inside the earth makes tunnel construction, shall we say, problematic?
The video says that it would take 42 minutes "regardless of the path you take".
This is obviously garbage. (To see why without math, consider the path that goes from LA to Tokyo... via another city.)
(In fact, it has to be a straight line path. (Assuming the various idealised conditions, of course.)
Outside of the frictionless tunnel it actually kinda works.
They didn't mention that the Earth would have to be perfectly spherical and uniform density. Also, there's coriolis forces if the tunnel isn't directly on the north and south pole.
It's just the History Channel, cut them some slack :P
The answer is _always_ 42.
Surely I can't be the _only_ reader of Douglas Adams around here who is profoundly DISTURBED by this?
Disturbed? Why, it makes perfect sense. How else would the Earth unit fulfil its proper function, simian?
If the cities are not antipodal, the time can always be made less than 42 minutes using a curved path. What curve gives the shortest time? This is the famous brachistochrone problem, which led to the development of the calculus of variations in the eighteenth century. If the cities are close enough that the curvature of the earth can be neglected, the optimal curve is a cycloid.
GrrlScientist, what bit most annoys you on this. Without doing the math myself, it seems like a perfectly typical 'ignore all the complicating realworld factors' toy problem. Yeah, they definitely should include the 'simplified case' caveat... but constructing a straight tunnel thousands of miles long through a flowing mantle (much less the core) already puts it out of the realm of 'realistic'.
Lots of mistakes, all down to the use of an oversimplified model - but this is an exercise in mathematics, not engineering, so I consider them acceptable simplifications.
Except for one - the ommision of rotational considerations. Chris pointed that out - any elevator would have a bumpy ride that involves a lot of crashing into the tunnel wall.
Conveniently for any occupents, the effect would actually provide them with some 'gravity' effect. I have no idea how much, but even a small amount of apparent gravity would make the trip more comfortable.