Special Relativity Calc: Half Speed of Light - Jose

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TL;DR Summary

Even applying the time dilation formula (Lorentz's equation) my calculations on Special Relativity seem to be wrong. the speed of light doesn't seem to be constant.

Hello!
I am trying to make some calculations on the Special theory of relativity over a practical example, considering the time dilation, but I may (more than likely..) be doing something wrong (probably with my assumptions). I would appreciate any comment on it.

The example is that of a rocket traveling at half the speed of light, so let´'s say at 150.000 km/s.
I am assuming that for those who are inside the rocket to measure the speed of light as that of 300.000 km/s (because it is a constant for all observers), this means that they see the light (its photons) 300 km ahead of the rocket, when a second (counted from inside the rocket) has passed (because, as relativity says, speed of a moving object related to another moving object is the difference between the two speeds). Light has traveled 300km more than the rocket (from the point of view of the guys inside the rocket) after a second.

Then my calculations are as follow:
. When a second within the rocket has passed, they have actually advanced 173 kms, since, due to the time dilation, that second is slower than in a stationary clock and is the equivalent to 1.1547 stationary-clock seconds (as a result of the time-dilation formula: t'= ϒ t, where ϒ would be 0.866 for a speed of 150km/s). So, traveling at 150 km/s during 1.1547 secs means you travel 173 kms in a second.
. And the light will reach 346,42 kms, going at 300 kms during 1.1547 secs.
But then, 346,42 - 173 is less than 300. How come the guys in the rocket measure a speed of light of 300km/s (or what seems to be the same: that they see the light at 473km, 300 ahead of them)??

Thank you in advance,
Regards,
Jose

 

[Nugatory]

You are not allowing for length contraction and the relativity of simultaneity.

You may be able to get a sense of where you are going wrong if you try analyzing this problem using a frame in which the rocket is at rest (and the "stationary" clock is moving backwards at 150000 km/s so is the time-dilated one). This is the exact same physical situation.

To really do it right though you will need to use the Lorentz transformations. Consider the events:
A) Flash of light leaves rocket when rocket clock reads zero and stationary clock reads zero.
B) Flash of light is 300 km ahead of rocket according to rocket measurements when rocket clock reads one second (because if constant speed of light).
C) Flash of light is 300 km in front of rocket according to stationary measurements and stationary clock reads one second (because of constant speed of light).

Use the Lorentz transformations to work out what the stationary clock reads at the same time that event B happens and where the flash is at that time. Then use them to work out what the rocket clock reads and where the flash of light is relative to the rocket at the same time as event C.

As an aside, you will find the arithmetic to be a bit easier if you:
1) Choose to measure time in seconds and distances in light-seconds so that the speed of light is exactly 1 (one light-second per second) instead of 150000 (km per second).
2) Choose the rocket speed to be .8 light-seconds per second... this makes all the square roots in the Lorentz transformations come out to even numbers.