Velocities in Special Relativity

Relative Velocities at Low Speeds - Easy!

Imagine a situation in which two cars are on a direct collision course towards each other, both travelling at 50 mph. What is the velocity of the cars relative to each other? I.e. if you were sitting in the first car, how fast would the second car be speeding towards you? Usually this is an easy question to answer: the relative velocity is the second car's velocity minus the first car's velocity. Relative velocity = 50 mph - (-50 mph) = 100 mph

Relative Velocities at High Speeds - Not So Easy...

However, when the vehicles are travelling at close to light-speed (186282 miles per second), it's not so straightforward. This is because of Einstein's theory of special relativity, which says that all observers, no matter how fast they are travelling, must observe the speed of light to be 186282 miles per second. No matter how fast you run after a beam of light, it still appears to be travelling away from you at 186282 miles per second.

This is only possible if the definitions of "miles" and "seconds" change. What looks like a mile to the person travelling at high speed actually looks like much less to a person who is standing still. This is called length contraction. Similarly, what feels like a second to the person travelling at high speed actually feels like much more to a person who is standing still. This is called time dilation.

Velocity Composition Formula

In order to take account of this, you need to apply a correction factor to the calculation of relative velocity.

Instead of the simple formula that applies at low speeds:

Relative Velocity = (v1 - v2) (where v1 and v2 are the velocities of the first and second objects)

In special relativity, we need to use the formula:

Relative Velocity = Γ﻿ (v1-v2)

where the correction factor Γ﻿ = 1/(1+v1*v2/c^2) and c is the speed of light.

Example Calculation

A ------------> <-------------B

Consider two particles, A and B, on a collision course. Each is travelling at 75% of the speed of light. Calculate their relative velocity as follows:

v1 = 0.75 c, v2 = -0.75 c

Relative velocity = (0.75c - (-0.75c) ) / (1+ 0.75^2 c^2/c^2) = 1.5 c / (1+0.75^2) = 0.96 c

Note that if you used the non-relativistic formula, you would get the incorrect answer 1.5 c - this is clearly wrong because nothing can travel faster than the speed of light!

topquark works as a researcher in theoretical particle physics and blogs about research at The Particle Pen.

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quicksand 5 years ago

This answers my question clearly. Thanks very much, TopQuark!

By the way, I have another hypothetical question.

If you are travelling on a straight path AB at the speed of light (which is impossible because mass will increase to infinity) and if you flash a beam of light in the direction of your motion, how would you, the observer be able to percieve that beam of light?

Thanks again TopQuark.

wandererh 5 years ago from Singapore

Hi quicksand, I want to know the answer to that one as well. :)

Anyway, sorry for posting here but I just want to point out that there is a typo in the figure for the speed of light in the second paragraph. :)

topquark 5 years ago from UK Author

Good question. If you were travelling at the speed of light, you'd experience infinite time dilation - i.e. time would stand still for you... so it's hard to imagine seeing anything.

If you were at 99.999999% of the speed of light, you would see what you usually do when you shine a beam of light away from you - the light would reflect off objects/dust and bounce back into your eye so you would see it. However, the light would be massively blue-shifted because the objects it reflects off are travelling towards you at high speed.

Blue shift: http://en.wikipedia.org/wiki/Blueshift

Spirit Whisperer 5 years ago from Isle of Man

You explain this beautifully. Do you have a simple way of explaining the the Lorentz Factor and its derivation?

topquark 5 years ago from UK Author

No, but I should do! I'll put it on my list of hubs to write.

randomperson 4 years ago

This explanation solved all my cuirrent (physiscs) problems because it considered two particles on a collison course, thank you topquark

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