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Testing the Addition of Velocities Formula

Testing the Addition of Velocities Formula - velocity v to...

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Testing the Addition of Velocities Formula Actually, the first test of the addition of velocities formula was carried out in the 1850s! Two French physicists, Fizeau and Foucault, measured the speed of light in water, and found it to be c / n , where n is the refractive index of water, about 1.33. (This was the result predicted by the wave theory of light.) They then measured the speed of light (relative to the ground) in moving water, by sending light down a long pipe with water flowing through it at speed v . They discovered that the speed relative to the ground was not just v + c / n , but had an extra term, v + c / n - v / n 2 . Their (incorrect) explanation was that the light was a complicated combination of waves in the water and waves in the aether, and the moving water was only partially dragging the aether along with it, so the light didn’t get the full speed v of the water added to its original speed c / n . The true explanation of the extra term is much simpler: velocities don’t simply add. To add the
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Unformatted text preview: velocity v to the velocity c / n , we must use the addition of velocities formula above, which gives the light velocity relative to the ground to be: ( v + c / n )/(1 + v / nc ) Now, v is much smaller than c or c / n , so 1/(1 + v / nc ) can be written as (1 - v / nc ), giving: ( v + c / n )(1 - v / nc ) Multiplying this out gives v + c / n- v / n 2-v / n × v / c , and the last term is smaller than v by a factor v / c , so is clearly negligible. Therefore, the 1850 experiment looking for “aether drag” in fact confirms the relativistic addition of velocities formula! Of course, there are many other confirmations. For example, any velocity added to c still gives c . Also, it indicates that the speed of light is a speed limit for all objects, a topic we shall examine more carefully in the next lecture...
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