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I was kind of puzzled yesterday when thinking about this.

  1. If we observe an object moving away from us with 10m/s we would say the object is moving away from us really slowly
  2. But if we (humans, making an observation) would be the size of a proton, previously mentioned 10m/s would appear enormously faster than in the first case

What puzzles me is this. If we say two protons move away one of another with relative speed 0.999c, then those (second case) two-man-protons would say the relative speed between them is much greater than 0.999c.

Does this make any sense?

Qmechanic
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sabiland
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3 Answers3

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If I understand you correctly, your two points about apparent slowness of speeds is related to scale, and disappears when you quantify it using a common unit.

ie: We think of 10m/s as relatively slow because the average human is 1.8 metres in height, and we can imagine that 10 metres per second, or 36 kilometer/hour as an achievable speed using a machine (car).

If we were the size of a proton, obviously, 36 km/h would appear to be quite fast. It's quite fast even if we were the size of ants.

But in physics, we quantify everything. To human of our size, AND to a human the size of a proton, we will have to agree on scales and units, and thus the speed in itself will not change due to the size consideration alone.

As for you final scenario: the relative speed in both cases is 0.999c, and c would be the same unit for both parties, so the relative speed is the same. On the other hand, if we used the height of human and diameter of proton(assuming hard spherical shape for simplicity) covered in a second as the unit in each case respectively, then we would have a different numerical value as the units are different. If the numerical values were the same, then the speeds would actually be different because the units are different.

mehfoos
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The important thing is that all speeds seem to change:

If you would look at the universe, in this case the two protons, from the perception of a man shrunk to the size of a proton, not only would the particles appear much faster, but so would the speed of light.

So if you shrunk yourself to a trillionth of what you are now, one proton would have a diameter of 40 centimeters, or 15.7 inches, relative to you. The speed of light, then, would also a billion times greater, again, relative to you. In 'absolute' terms, i.e. relative something that did not change its size, nothing would change.
In your terms, the speed of the protons would go from 0.999c to 999,000,000,000c but the speed of light would have gone up, too: from c to 1,000,000,000,000c. Now if you look at the numbers, you can see that the ratio staid the same: 0.999.

Hope I could help.

Sincerely Marrus with a Q

Marrus Q
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No this doesn't make sense.
Your argument number $2$ is wrong because if we(humans making the observation) shrink to the size of proton then we would use metre scales of size of order of our(human's) size whose one unit would be much smaller than 1 metre. The magnitude of the measurement made by this small scale would be larger in numbers but then to compare this measurement with $c$ we would have to convert the units of our measurement to the units in which $c$ is measured .
More precisely let $v$ be the velocity of that object then the ratio $\dfrac{v}{c}$ will be less than unity if $v$ and $c$ are measured in same units.

user31782
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