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EnlightenedOne
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Homework Statement
You shine a powerful laser onto to the surface of the Moon from Earth (Earth-Moon distance is 384,000 km or 3.84E8 m). About how fast must the laser pointer rotate (in degrees per second) for the spot on the Moon to move with velocity v>c? Does this violate Special Relativity?
Homework Equations
No equations were given.
The Attempt at a Solution
This problem is in the Special Relativity section of a Modern Physics class. At first glance, I had absolutely no idea where to start. My professor didn't give me any equations for this type of problem. So, here is my bad attempt (probably wrong):
I started by thinking of a relation between the angle (θ) of each incremental rotation and the distance (d) the spot moves on the Moon. If you make the distance from Earth to the Moon the horizontal, and you rotate the laser at an angle θ from the horizontal, you get a right triangle (assuming the Moon's surface is flat) with opposite=d and adjacent=Earth-Moon distance. So, if you take the tangent of the angle, you get:
tanθ = d/(3.84E8)
So the distance the spot moves on the Moon after a rotation θ is:
d = (3.84E8)tanθ
I then assumed that the velocity of the spot on the Moon is given by:
v = d/t, where t is the time the spot takes to move that distance (or, equivalently, the time it takes for the laser to rotate by θ)
At this point I calculated that the time it takes for the light to move from the laser pointer to the Moon is:
t = (Earth-Moon distance)/(speed of light) = (3.84E8)/(3E8) = 1.28 s
If you make the time interval between each incremental rotation 1.28 s, then you get the following equation for the velocity of the spot:
v = d/t = ((3.84E8)tanθ)/(1.28) = c*tanθ
If you want v>c, then:
c*tanθ>c
tanθ>1
θ> 45°
But, I don't know how to interpret my answer (i.e. 45°/s or 45°/1.28s).
More than that though, my method is probably completely wrong anyway and I need help.
Can someone please show me how to do this problem? I'm completely lost.
Thank you