Rotating a laser at Moon so that spot has v>c

[EnlightenedOne]

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

 

[happysmiles36]

I'll explain the "does this violate special relativity?" first.

It doesn't because the laser is ejecting 1 proton at a time, this happens so fast it seems like a continuous beam when you move it. However if the angles of 2 photons are different over a long distance, they can end up being far apart. So its like 2 different events.
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This is how I would set it up, kind of a weird question because it is impossible for there to be a answer because the photon travels at the speed of light and can't go faster and if you have a right triangle with a photon on the side and the hypotenuse traveling with the same speed, they wouldn't meet up and the side photon would always be covering more distance. And the spot on the moon is going to be stationary because that is only one photon and in another sense, light moving with only a x direction will always be more than light with a x and a y direction. BUT I will assume the speed for the laser to travel is instantaneous. Maybe my logic is flawed as well haha.:tongue:I'm going to say for the first part to get the speed to travel faster than light, the distance between the 2 points has to be greater than 3.0x10⁸m. The distance from Earth to the moon is 384,000,000m. So we draw a right triangle. The y being distance from Earth to Moon and the x being 3.0x10⁸m distance.

From this you can find the angle and it will already be in degree/second because we used 3.0x10⁸m (c*1s).

However this value (degree/s) is equal to the speed of light, therefore you need to indicate that it has to be greater than this value.

 

[EnlightenedOne]

I'll explain the "does this violate special relativity?" first.

It doesn't because the laser is ejecting 1 proton at a time, this happens so fast it seems like a continuous beam when you move it. However if the angles of 2 photons are different over a long distance, they can end up being far apart. So its like 2 different events.
---------------------
This is how I would set it up, kind of a weird question because it is impossible for there to be a answer because the photon travels at the speed of light and can't go faster and if you have a right triangle with a photon on the side and the hypotenuse traveling with the same speed, they wouldn't meet up and the side photon would always be covering more distance. And the spot on the moon is going to be stationary because that is only one photon and in another sense, light moving with only a x direction will always be more than light with a x and a y direction. BUT I will assume the speed for the laser to travel is instantaneous. Maybe my logic is flawed as well haha.:tongue:


I'm going to say for the first part to get the speed to travel faster than light, the distance between the 2 points has to be greater than 3.0x10⁸m. The distance from Earth to the moon is 384,000,000m. So we draw a right triangle. The y being distance from Earth to Moon and the x being 3.0x10⁸m distance.

From this you can find the angle and it will already be in degree/second because we used 3.0x10⁸m (c*1s).

However this value (degree/s) is equal to the speed of light, therefore you need to indicate that it has to be greater than this value.

I agree, this question is very weird:P.

Oh, ok! When I do what you said, I now get a rotation speed of 37.926°/s (for it to equal light speed) so anything higher does yield a speed greater than light. Thank you for clarifying this! My previous answers were 45°/s and 35°/s but the first was too high and the second was too low, and so it makes sense for it to be 37.926°/s.

Its always the small stuff that brings me so much confusion. I sure hope its right:D! I really hate it when professors don't even talk about this stuff and then expect us to figure it out.

I really appreciate the explanation:D!

Thanks again!

 

[happysmiles36]

Glad it helped :D

 

[HalfLostAndSearching]

for your help!

As a scientist, it is important to approach problems with a critical and analytical mindset. In this scenario, we must consider the laws of physics and the principles of Special Relativity.

Firstly, it is important to note that the speed of light is the ultimate speed limit in the universe. According to Special Relativity, the speed of light (c) is constant for all observers, regardless of their relative motion. This means that nothing can travel faster than the speed of light.

In this problem, we are asked to rotate a laser at a speed that would result in the spot on the Moon moving faster than the speed of light. This raises some red flags and suggests that something may be wrong with our approach or assumptions.

One potential issue with the given problem is the assumption that the distance between the Earth and the Moon is the same as the distance between the laser pointer and the spot on the Moon. This is not entirely accurate, as the laser beam will spread out and cover a larger area on the Moon's surface. This means that the spot will not be moving at the same speed as the laser pointer, as it is covering a larger distance in the same amount of time.

Additionally, our approach of using trigonometry to calculate the velocity of the spot may not be valid in this scenario. This is because Special Relativity states that time and space are relative and can be affected by an observer's frame of reference. As we are dealing with moving objects and velocities close to the speed of light, the effects of Special Relativity cannot be ignored.

To truly determine the velocity of the spot on the Moon, we would need to use the principles of Special Relativity and the concept of relativistic velocity addition. This would take into account the relative velocities of the laser beam, the Earth, and the Moon, and would give us a more accurate and valid answer.

In conclusion, while the given approach may give us a rough estimate of the required rotation speed of the laser, it is not a scientifically rigorous solution. To accurately determine the velocity of the spot on the Moon, we would need to use the principles of Special Relativity. Therefore, it is not appropriate to say that we can rotate a laser at a speed that would result in the spot on the Moon moving faster than the speed of light. This would violate the fundamental principles of Special Relativity.