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A mirror is placed 1 light hour away from a powerful laser pointer. Right when the laser is turned on, Alice move from laser pointer towards the mirror at 1 m/s. Bob stays where the laser pointer is.To make this simple, we will assume Alice is already moving towards the mirror at 1 m/s relative to the pointer as she passes it and it is turned on.
Who will see the reflected laser first?
What's the speed of light before and after reflection when measured by Alice? Is it still the same as distance divided by travel time?
Because she is moving at 1m/sec relative to both pointer and mirror, she would not measure the distance between them as being 1 light hr, but rather 0.99999999999999999443674971973191 light hr. as measured relative to herself, the laser will travel at c, towards the the mirror, while the mirror moves towards her at 1m/sec. It will take 0.99999999666435905358172976595532 hrs for the laser and mirror to meet. During which time the distance between Alice and the mirror would decrease by 3599.9999879916925928942271574392 meters. The laser will then return at c from this distance to Alice. At this point, I'm going to quit giving numbers, because it is just too cumbersome to keep working with all these decimal places.
Instead, I'm going to change the scenario to use a velocity for Alice that is easier to work with.
Thus Alice now moves at 0.6c. relative to laser pointer and mirror.
As Alice passes the pointer, she will measure the distance to the mirror as being 0.8 light hrs, and the mirror will be approaching at 0.6c, while the light from the laser will be speeding towards it at c relative to Alice. Thus, according to Alice, the light will take (0.8 lh/(1c+.6c) = 0.5 light hr to meet up with the mirror. During which time, the distance between Alice and the Mirror will have decrease by 0.5 hr * 0.6c = 0.3 lh to 0.5 light hr. The refelcted light, traveling at c will take 0.5 hr to get back to Alice. Total time for Alice for round trip of laser 1 light hr.
Now, if we work it out from Bob's view:
The light travels at c towards the Mirror, arrives at the mirror in one hour and returns 1 hr later, total trip time 2 hr.
Alice, is moving towards the mirror at 0.6 c, so, in the time it take for the light to reach the Mirror, Alice will have moved to be 0.6 light hrs closer to the mirror to be 0.4 lh away from it, and will meet with the returning light in another (0.4 lh/(1c+0.c) = 0.25 hr, total time between firing of laser and it meeting up again with Alice is 1.25 hrs. However, since Alice is moving at 0.6 c relative to Bob, He will measure her clock as time dilated by a factor of 0.8 and only tick off 1.25hr * 0.8 = 1 hr. The same amount of time that Alice says here clock ticked off.
If we go back to Alice's view, we can also work out how much time she would say ticks off on Bob's clock between Firing the laser and the light returning to Bob.
As noted above, Alice measures 1 hr for the light to meet up with her. During which time, she would measure Bob's clock as being time dilated and ticking off 0.8 hr. Now in that hr, Bob, with a relative velocity of 0.6c, has moved 0.6 lh away from Alice. The light passing Alice on it way to him has to "chase after" the receding Bob at c. This takes 0.6 lh (1-0.6c) = 1.5 hours by Alice's clock. During which time Bob's clock accumulates 1.5 hr *.8 = 1.2 hr. Added to the 0.8 hr already accumulated equals 2 hrs total time accumulated by Bob's clock according to Alice. The same as what Bob's recorded.
Both Alice and Bob agree that the reflected Laser hits Alice first. ( though Alice would say that when the Light hits her, Bob's clock read 0.8 hr, and Bob would say that when the light hit Alice his clock read 1.25 hrs, And Alice would say that when the light returns to Bob, her clock reads 2.5 hrs, while Bob would say that her clock reads 1.6 hrs.
So the answer to what the speed of the light is according to Alice both before and after refection, it is c ( relative to Alice.) just like it is c relative to Bob as measured by Bob's. It is also distance divided by travel time, keeping in mind that Bob and Alice do not measure either distance or time the same.
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