If two trains traveling at 0.5c collide head on

[CombustionMan]

What would happen? These two trains are on the same tracks moving at constant velocities towards each other (0.5c --> and <--0.5c I guess) so what would happen? Since all motion is relative and no mass can travel at the speed of light something else must happen instead of a light speed head-on collision, right? I'm just curious.

 

[Calrid]

The difficulty is in accepting that unlike at low velocities, where two objects moving at say 60mph are additive and so the relative velocity is 120mph, at speeds approaching c this is not the case.

The most a mass object could achieve would be something approaching a relative speed of 1c. There would be a very large explosion though and a lot of energy would be produced.

The Lorentz transforms explain it but essentially at significant fractions of light speed, the dilation of time and space means that relative velocities are always below 1c.

$\ \gamma = \frac{1}{ \sqrt{1 - { \frac{v^2}{c^2}}}}$

Which leads to the composition of velocities equation:

Where 2 moving bodies are w and v

Best way to imagine it is to try and understand what time and motion dilation means, distance and time are equivalently dilated by motions that approach significant fractions of c, so that the relative concerns mean that they are effectively traveling through time space that is becoming "stretched" and hence the relative motion of mass objects becomes more dilated as they near velocities approaching c until the dilation itself prevents anything with mass achieving this limit. Probably not a very good explanation but meh.

 

[CombustionMan]

I think I understand, so since they are traveling through "stretched" time their relative velocities go below 1c?

 

[HallsofIvy]

The formula for the relative speeds of two objects moving toward each other with speeds (relative to a third observer) of u and v i s NOT "u+ v". The speed of each relative to the other will be
$$\frac{u+ v}{1+ \frac{uv}{c^2}}$$

In particular, if $u= v= c/2$, then the speed of either one relative to the other is
$$\frac{\frac{c}{2}+ \frac{c}{2}}{1+ \frac{\frac{c}{2}\frac{c}{2}}{c^2}}= \frac{\frac{c}{2}}{1+ \frac{1}{4}}= \frac{4}{5}c$$

 

[Calrid]

I think I understand, so since they are traveling through "stretched" time their relative velocities go below 1c?

Yup pretty much although it's stretched time and space, space-time. You can play with Ivy's equations yourself to show that speeds always approach c just by plugging some numbers in.

It's more interesting if you take the case of .999c and then you would imagine the relative velocities are 1.998c which is clearly impossible but again just plugging the numbers in the dilation effects conserve c as a limit.

Interestingly we can observe this with GPS satellites, since the gravitation in space is slightly lower than on Earth and the relative speeds although far slower than c become significant over time. Without adjustments they can go out of synch fairly significantly in quite short periods of time. So to adjust for it they lower the frequency of clocks so that they take account of the respective dilation effects.

They have also placed Caesium clocks on tall buildings and flown them around the world to show extremely small variations that are predicted by the theory.