How to calculate the speed of anything, when their speed becomes closer to the speed-of-light.
This article was originally a comment in the linked article; Why Nothing Moves Faster Than Light.
— To correct the comment I have made earlier ” unless something is completely massless in its rest-frame ” I also add the following. This is a fact which I have realized lately — or rather trapped myself to commit an inconsistent remark, by following the same comment.
But it’s better late than never; when something is massless, it will never have a rest-frame, because by Einstein’s transformation rules known as Theory of Relativity to be consistent, a massless particle will always move at the speed of light, no matter which frame we are looking at it from.
This then leads to the velocity addition formula of Einstein.
According to Newton and Galileo — Galileo Project., known by a name Galilean Relativity, the following follows; if C moves at speed speed_cb wrt B, B moves at speed speed_ba wrt A then C must move at speed speed_ca given by speed_ca = speed_cb + speed_ba. The speeds get added up.
Let’s take this simple example, to illustrate the above; A bullet moves at speed 200 km/hr in a train and the train moves at 250 km/hr wrt Jason, who is standing still, on ground, then ground is A, train is B and bullet is C.
So Jason will see that the bullet will move at speed 200+250 = 450 km/hr. Thus Galilean Relativity: speed_ca = speed_cb + speed_ba. Note that the b as an index vanishes, if we keep the indices in proper sequence. ca = cb + ba
But Einstein modified this, to make the rule consistent, with how it will work out for light or any relay of information at that speed. To make light always move at speed c = 2.99 x 10 ^8 meter/sec the simple addition becomes “warped”.
It’s still known as addition of velocities or addition of speed and given by speed_ca = [ speed_cb + speed_ba ] / [1 + speed_cb * speed_ba] where we will get the results in terms of speed of light being 1 and speed of anything else as a fraction given by speed_ca.
We then simply multiply c = 2.99 x 10 ^8 meter/sec to that fraction, to get the exact speed, in terms of meter/second or km/hr etc. Not only we have to multiply this value to any result we obtain from this formula, we also have to divide this value 2.99 x 10 ^8 meter/sec before carrying out any calculation, in order to use the formula, given in this form.
— Literature will make it slightly more complicated, but less attractive, for modern Physicists, to get consistent results in meter/sec or km/hr.
So Einstein’s Relativity: speed_ca = [ speed_cb + speed_ba ] / [1 + speed_cb * speed_ba] .
When this fraction [1 + speed_cb * speed_ba] is divided, its no more a simple addition, but a warped addition and known as Einstein Velocity Addition Formula.
This article has a good deal of similitude of content with the linked: Why Nothing Moves Faster Than Light — so they can be read together.