Question

Hypothetical question, if a particle(A) traveling near the speed of light (99.99 ...%) and is compared to another particle(B) traveling at the speed of light (100%).
Would the particle B traveling past the particle A at a sub speed of light (the difference) or at the speed of light/

Answer 1

The speed of light is the same in all reference frames, so particle B would appear to be travelling at the speed of light to particle A. Let's look at a really fast space ship and a photon of light.

We're on the spaceship, going 99.99% the speed of light, and we measure some things about our photon. We measure the wavelength and the frequency. A person who isn't moving also measures these things.

The person on the ground finds that when they multiply the wavelength and the frequency, they get the speed of light, c, just as we'd expect.
\[c = \lambda \cdot f\]
On the spaceship, we actually find the same thing. The product of the wavelength and frequency equals the speed of light. One thing, however: we didn't measure the same wavelength and frequency as the stationary person.

The speed of light is a constant in all inertia reference frames, but the wavelength and frequency are dependent on our motion relative to the source of the photon.

Now, for particles travelling really fast that have mass, we can find out how fast they'd appear to be going to one another with this equation:
\[V_{relative} = \frac{u+v}{1+\frac{uv}{c^2}}\]
where u and v are the speeds of the two particles. This equation works fine if they are travelling parallel or anti-parallel to one another.

You'll notice that if you plugged in c for either u or v, the equation reduces to c. So if one particle is going the speed of light, it will appear to be going the speed of light to all observers in all inertial reference frames.