# A Roller Coaster Ride through Relativity/Relative Velocity

## Relative Velocity[edit]

Suppose my brother and I have two identical twin sister ships, each 100 m long when at rest. If we pass each other with a relative velocity of 80% of the velocity of light, to me my brother's ship appears to be only 60 m long. (g = 1.67 so l' = 100/1.67 = 60 m.)

If I time his passing with an accurate clock I will find that he takes 0.25 µs to go by. Using the formula speed = distance / time I calculate his speed to be 60/(0.25 x 10^{−6}) = 240,000,000 ms^{−1}.

Of course, if my brother makes the same measurements on my ship he will come to exactly the same conclusion about my speed.

But what does *he* think of *my* measurements and I of his? Lets concentrate on the latter. I watch as he approaches my ship and observe him start his clock at the instant he passes the front of my ship and stop his clock at the instant he passes the back. His clock stops at 0.25 µs. So far so good.

I observe him doing his calculations. He starts to press the 6 and the 0 buttons. 'No, that's wrong!' I exclaim. 'My ship is not 60 m long it is 100 m long!' Then I watch him divide by the time 0.25. 'No, that's wrong too,' I exclaim. 'Your clocks are running slow! You should be dividing by 0.417 not 0.25!' (0.417 is of course 0.25 x 1.67) Grabbing a calculator I perform the calculation for him - only to get... 100/(0.417 x 10^{−6}) = 240,000,000 ms^{−1}! So it doesn't matter who does the calculations, we still get the same result. Each of us is convinced that the other has used the wrong data but we both agree about the answer!

## The big loop[edit]

*Well, I am glad about that!*

As I glance ahead I see that the roller coaster is hurtling towards one of those up-and-over loops.

*But there is still something puzzling me about the penny.* you say. *How can both ends of the penny start to fall at the same time in one frame of reference while the flea sees the front start to fall before the back? I don't get it.*

Over on the left, you hear the sound of a train. This one is really shifting and you watch it plunge into the tunnel. The train you saw before had just the same number of carriages and was the same length as the tunnel but you are not surprised now to see that this one seems a lot shorter and there is no sign of the engine emerging when the last carriage plunges out of sight.

Suddenly you hear a couple of loud explosions. Two large clouds of dust rise from the two ends of the tunnel on the railway line. Someone – a terrorist perhaps – has blown up the tunnel!! A moment later, the engine of the train bursts through the heap of rubble at the exit of the tunnel and you watch horrified as the coach after coach piles into the wreckage.

*Look at that! The whole train has been completely wrecked!*

Yes: it was going so fast, it was shorter than the tunnel.

*Yes, but – surely that's just a kind of illusion isn't it? The train is really exactly the same length as the tunnel so, it stands to reason that, at the instant the last carriage entered the tunnel, the engine must emerge from it. Rulers might shrink and clocks run slow – but surely, if two things happen at the same time – they, well, happen simultaneously, don't they?*

Not necessarily. Basically, the whole notion of simultaneity has to be abandoned and we must accept the brutal fact that:

Bizarre consequence number 5 |
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Events which occur at different places but at the same time to one observer may happen at different times according to another. |

But just at that moment your world begins to turn upside-down…