Firstly, I apologize if this is the wrong place to post this question; I'm not a scientist, just curious. So: if the speed of light (299 792 458 m/s according to Google) is an absolute limit even for relative speeds, then what happens if I throw a rock such that it achieves the speed of light, and then throw another in exactly the opposite direction (at any speed I guess, but lets say speed of light)? What is their velocity relative to one another?
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"is an absolute limit even for relative speeds" - what do you mean by relative speeds? – Nemo Nov 06 '17 at 10:32
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well, if I throw two rocks in opposite directions at, say, 1 m/s then the speed of either rock relative to the other is 2 m/s. I meant it in that way. – Numi Nov 06 '17 at 10:34
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If you want to calculate the relative velocity between two objects, you have to use composition law for velocities:
$$v=\frac{u+w}{1+\frac{uw}{c^2}}$$ where $c$ is the speed of light.
Even if the two objects travel at the speed of light (which is not possible for a massive particle) the relative velocity cannot exceed $c$.
$$v=\frac{c+c}{1+\frac{c^2}{c^2}}=c$$
Note that for low velocities, the term $\frac{uw}{c^2}\rightarrow 0$, so that $1+\frac{uw}{c^2}\rightarrow 1$. In the approximation of low velocities, the composition law for velocities reduces to:
$$v= u+w $$
Nemo
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Ok fair enough, but I'm having extreme difficulty visualizing how this works: relative to me (the observer who threw both rocks, standing at say the origin), the rocks are going at light speed relative to each other AND relative to me? – Numi Nov 06 '17 at 10:42
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Is this where the whole time dilation thing comes into play? Please suggest something that I can read to gain a greater understanding of this. – Numi Nov 06 '17 at 10:43
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@Numi, Yes, the speed of light is $c$ relative to every inertial frame of reference. The speed of light is the same in every inertial frame of reference. This is one of the postulates of special relativity. – Nemo Nov 06 '17 at 10:45
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@Numi, Yes, this is the reason why space and time are not invariants. They form an invariant quantity called spacetime interval. But separately they are not invariants. – Nemo Nov 06 '17 at 10:49
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The second equation is an invalid application of the velocity addition formula since, in its derivation, one of the speeds is the relative speed of two inertial reference frames and that cannot be $c$. – Alfred Centauri Nov 06 '17 at 12:02