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Moved from a technical forum, so homework template missing

I'm working through an introductory book on special relativity (by Resnick and Halliday), and am having trouble with one of the end-of-chapter problems.

Sue and Jim are two experimenters at rest with respect to one another at different points in space. They “fire” neutrons at each other, each neutron leaving its “gun” at a relative speed of 0.6c. Jim fires neutrons at a steady rate of 10,000 neutrons/second. State the rate that would be reported by a 3rd person (Bill) who is in a frame chosen so that Sue’s neutrons are at rest in it.

One or more of: Lorentz transformation, relativistic velocity addition, relativistic doppler effect.

Define frame S as that of Jim and Sue.

Define frame S’ as that of Bill.

Then in S, the neutrons are fired from the same location and at times separated by 100μs. So

Δt = 100μs and Δx = 0.

Using the Lorentz transformations:

Δt’ = γ(100μs) = 125μs.

But this means that the frequency as seen by Bill is 8000/s. But the book says 12,500/s, and I’m sure it’s correct. After all, when a source approaches you, the frequency of anything it “throws” or radiates towards you is higher than if it remains motionless with respect to you.

I suspect that the problem is similar to that of a researcher on earth who is measuring the frequency of radiation coming at earth from elsewhere. The frequency will be higher if the object radiating is approaching earth. But I don’t think it’s simply a Doppler effect (besides, if I use the Doppler effect equation, I still don’t get the answer).

Any help appreciated.

**Problem Statement:**Sue and Jim are two experimenters at rest with respect to one another at different points in space. They “fire” neutrons at each other, each neutron leaving its “gun” at a relative speed of 0.6c. Jim fires neutrons at a steady rate of 10,000 neutrons/second. State the rate that would be reported by a 3rd person (Bill) who is in a frame chosen so that Sue’s neutrons are at rest in it.

**Relevant Equations:**One or more of: Lorentz transformation, relativistic velocity addition, relativistic doppler effect.

**Attempt at a Solution:**Define frame S as that of Jim and Sue.

Define frame S’ as that of Bill.

Then in S, the neutrons are fired from the same location and at times separated by 100μs. So

Δt = 100μs and Δx = 0.

Using the Lorentz transformations:

Δt’ = γ(100μs) = 125μs.

But this means that the frequency as seen by Bill is 8000/s. But the book says 12,500/s, and I’m sure it’s correct. After all, when a source approaches you, the frequency of anything it “throws” or radiates towards you is higher than if it remains motionless with respect to you.

I suspect that the problem is similar to that of a researcher on earth who is measuring the frequency of radiation coming at earth from elsewhere. The frequency will be higher if the object radiating is approaching earth. But I don’t think it’s simply a Doppler effect (besides, if I use the Doppler effect equation, I still don’t get the answer).

Any help appreciated.