Chupacabra wrote:...with a twist. Obviously very unrealistic, but a good theory question.
A train is moving 3/4ths the velocity of light (c = 300,000 km/sec) relative to a lamp post. Assume that a person is running 3/4ths the velocity of light on top of the train.
What is the velocity of the person relative to the lamp post?
edit: the person is running 3/4ths the velocity of light relative to the train.
This question was put forth by a famous physicist named George Gamow. You might have heard of him.
Anyway, the correct answer of course is based on the fact that he can't move faster than the speed of light. But why or more specifically what is his velocity is a bit more tricky. The right answer is (24/25)c.
The basic idea as far as I understand it is that the classical vector addition theorem is wrong. It works for most practical purposes, but it isnt 100% correct. And the larger the numbers you're dealing with, the more off you get.
In other words, any time you want to combine (for lack of a better term) two velocities, you can't just add them up. Two velocities cannot simply be added together. An explanation below is provided.
In order to explain best I'm just going to quote George Gamow from his book: "Mr Tompkins in Wonderland"
According to the theorem of addition the total velocity should be one and half times that of light, and the running tramp should be able to overtake the beam of light from a signal lamp. The truth, however, is that since the constancy of the velocity of light is an experimental fact, the resulting velocity in our case must be smaller than we expect--it cannot surpass the critical value c; and thus we come to the conclusion that, for smaller velocities also, the classical theorem of addition must be wrong.
The mathematical treatment of the problem, into which I do not want to enter here, leads to a very simple new formula for the calculation of the resulting velocity of two superimposed motions.
if v1 and v2 are the two velocities to be added, the resulting velocity comes out to be:
V = (v1 +/- v2)/(1 +/- ((v1)(v2))/c^2)
(er in case my brackets aren't right and to get a clearer picture, I drew it in Paint:)
(he goes on to say that for small velocities (compared to the speed of light), you can neglect the second term in the denominator for most practical purposes)
In a particular case, when one of the original velocities is c, the forumula gives c for the resulting velocity independent of what the second velocity may be. Thus, by overlapping any number of velocities, we can never surpass the velocity of light.
You might also be interested to know that this forumla has been proved experimentally and it was really found that the resultant of two velocities is always somewhat ersmaller than their arithmetical sum.
ta da
