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The title isn't quite clear, because the question was a little too long. Here it is in full:

How to apply length contraction and time dilation to a moving observer to measure the speed of light of a light beam moving parallel to their reference frame.

Imagery and familiarity are my learning tools.

A vessel, whose pilot is named Cleopatra, is travelling at 2/3c. At an arbitrary moment and location, Cleo fires a beam of light from the nose of the ship. This event defines t=0 and x=0 for a nearby stationary observer, Anthoney. This event also defines t(prime)=0 and x(prime)=0 for Cleopatra.

At t=1 for Anthoney, he sees that Cleopatra has travelled 200000km and the light beam has travelled 300000km. He also sees that the time according to Cleo is t/(gamma). where gamma is approx. 1/0.74 approx. 1.34.

My current understanding is that, removing time dilation and length contraction, at this moment, t=1, Cleo will see that light has travelled just 100000km beyond the nose of the ship. Further, that by some process of length contraction and time dilation, this distance of 100000km can be adjusted such that when divided by the dilated time (0.74s) we will get some ratio of c. Specifically, it will need to be in the region of 223000km. Finally, that the process here is no different than if a beam of light had been emitted in the direction of motion at x=0 and t=0, by some 3rd party. i.e. we are only interested in the portion of the beam that has past the nose of the ship after any given time.

The first check is whether or not this understanding of mine is in fact correct.

I created a minkowski diagram and a light clock diagram for v=2/3c. The light clock I find to be more intuitive, despite the use of a clock 300000km long. However, I'm no closer to revealling the means of resolving the original question/statement.

How to apply length contraction and time dilation to a moving observer to measure the speed of light of a light beam moving parallel to their reference frame.

Imagery and familiarity are my learning tools.

A vessel, whose pilot is named Cleopatra, is travelling at 2/3c. At an arbitrary moment and location, Cleo fires a beam of light from the nose of the ship. This event defines t=0 and x=0 for a nearby stationary observer, Anthoney. This event also defines t(prime)=0 and x(prime)=0 for Cleopatra.

At t=1 for Anthoney, he sees that Cleopatra has travelled 200000km and the light beam has travelled 300000km. He also sees that the time according to Cleo is t/(gamma). where gamma is approx. 1/0.74 approx. 1.34.

My current understanding is that, removing time dilation and length contraction, at this moment, t=1, Cleo will see that light has travelled just 100000km beyond the nose of the ship. Further, that by some process of length contraction and time dilation, this distance of 100000km can be adjusted such that when divided by the dilated time (0.74s) we will get some ratio of c. Specifically, it will need to be in the region of 223000km. Finally, that the process here is no different than if a beam of light had been emitted in the direction of motion at x=0 and t=0, by some 3rd party. i.e. we are only interested in the portion of the beam that has past the nose of the ship after any given time.

The first check is whether or not this understanding of mine is in fact correct.

I created a minkowski diagram and a light clock diagram for v=2/3c. The light clock I find to be more intuitive, despite the use of a clock 300000km long. However, I'm no closer to revealling the means of resolving the original question/statement.

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