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Let's say I fire a bus through space at (almost) the speed of light. If I'm inside the bus (sitting on the back seat) and I run up the aisle of the bus will I in fact be traveling faster than the speed of light? Relative to earth that I just took off from.

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Also note that a bus could never travel at the speed of light. –  David Z Mar 23 '11 at 16:48
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Not to mention what would happen if you would turn on the headlights... –  Stein Åsmul Jun 23 '11 at 1:23
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No. Relative to Earth your bus will have zero length, so moving from back to the front of the bus will contribute nothing to your speed relative to Earth. –  Anixx Aug 15 '11 at 13:58
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@Inceptio: Actually, the lights will still shine in front of you. One of the postulates of SR is that the speed of light is fixed for all reference frames. –  LastStar007 Jan 11 at 23:39
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@DavidZ yeah, busses can barely travel half the posted speed limit. –  Nick T Feb 19 at 0:01

4 Answers 4

up vote 47 down vote accepted

Your question has to do with addition of velocities in special relativity. For objects moving at low speeds, your intuition is correct: say the bus move at speed $v$ relative to earth, and you run at speed $u$ on the bus, then the combined speed is simply $u+v$.

But, when objects start to move fast, this is not quite the way things work. The reason is that time measurements start depending on the observer as well, so the way you measure time is just a bit different from the way it is measured on the bus, or on earth. Taking this into account, your speed compared to the earth will be $\frac{u+v}{1+ uv/c^2}$. where $c$ is the speed of light. This formula is derived from special relativity.

Some comments on this formula provide direct answer to your question:

  1. If both speeds are small compared with the speed of light, they approximately add up as your intuition tells you.

  2. If one of the speeds is the speed of light $c$, you can see that adding any other speed to it does not in fact change it: the speed of light is the same in all reference frames.

  3. If you add up any two speeds below $c$, you end up still below the speed of light. So, any material object which has a mass (unlike light, which doesn't), moves at a speed less than $c$. Adding to it according to the correct rule makes it closer to the speed of light, but you can never exceed it, or in fact not even reach it.

I'd recommend Wheeler and Taylor's "Spacetime Physics" to read about this. Unlike the reputation of the subject it is actually pretty intuitive (I learned that formula in high school).

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holy crap, awesome. –  ed209 Mar 23 '11 at 13:54
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Great answer. I second the recommendation of Spacetime Physics. –  Ted Bunn Mar 23 '11 at 14:07
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I heard that space can expand faster than the speed of light. so what about if I am on a planet and you are on another, at opposite ends of the universe. We'd be waving goodbye to each other and disappearing into the distance at faster than the speed of light (although we'd never actually see each other as the light is not fast enough to reach each other) –  ed209 Mar 24 '11 at 14:13
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That's true: once space is stretching, instead of material things moving, more general things can happen. This is the subject of general relativity, which combines the finite speed of light (but only "locally") with gravity. There are still rules, they are just not as simple. –  user566 Mar 24 '11 at 14:59
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+1 Nice answer here –  Ebenezer Sklivvze Apr 3 '11 at 16:53

No. Relative to Earth your bus will have zero length, so moving from back to the front of the bus will contribute nothing to your speed relative to Earth.

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Doesnt the bus then have insane mass density? Why doesnt it collapse into a black hole from the observers reference? –  Skúli Mar 16 at 22:29
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I wonder how much Cherenkov radiation that would cause. –  Cees Timmerman May 7 at 8:59

I will have to answer this quickly, for I suspect this question will be closed. However, this thought experiment is similar to what Einstein thought about 10 years before he published his paper on special relativity. The problem is this. If you were on a reference frame moving at the speed of light you would observe that light, or any electromagnetic wave, as a wave of oscillating electric and magnetic fields. However, this would be stationary, which contradicts the Maxwell equations for the propagation of electromagnetic radiation. Einstein worked to fix this contradiction, which lead to special relativity. The conclusion is that you can’t place yourself on a frame where light is observed to have any velocity other than the speed of light $c~\simeq~300,000km/sec$.

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Why did you think this would be closed? (In general, if a question should be closed, flag it, don't answer it) –  David Z Mar 23 '11 at 16:47
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Then what about slow light? –  Cees Timmerman May 7 at 8:41
    
@Lawrence, You didn't actually answer the question... –  Pacerier Jul 10 at 16:15

Hmm.

Assume that your bus is approaching the speed of light, because if it had reached it, its mass would be infinite and the question becomes metaphysical as far as the contents and passengers.

Generally, momentum conservation insures that the bus would drop back from the speed it has to compensate for your momentum,as long as you are airborn but when you hit the front glass, it will gain it back.

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so if I'm floating inside the bus and I have a rocket pack on my back and fire it, how would the bus's momentum be affected? I'm not touching the bus. –  ed209 Mar 23 '11 at 13:51
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The gas from your rocket will hit the back wall of the bus and transfer the momentum. –  anna v Mar 23 '11 at 15:11
    
@annav You could float next to the bus and fire your rocket to pass it. I'd be worried about obstacles and oncoming traffic, though. –  Cees Timmerman May 7 at 8:50

protected by David Z Dec 17 '11 at 22:15

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