Basic relativity problem - Lorentz Transformations

[Akhilleus]

Homework Statement

Event A occurs at x_A = 500m. Event B occurs 5 microseconds later at x_B = 1500m. With what speed must an observer move in the positive x direction so that the events occur at the same point in space in the observer's frame?

Homework Equations

Lorentz transformation equations:
x' = g(x - Vt)
t' = g(t - (Vx)/c²)
g = 1/sqrt(1 - V²/c²)

The Attempt at a Solution

I understand conceptually that, as the observer approaches c, the distance between the two events will contract and the time will "slow down", but I'm unsure how to find these values. Is the V value in the above equations simply the distance divided by time when the events are observed at rest? If so, solving for x' leads to a value of 0. Any help would be greatly appreciated.

 

[tiny-tim]

Hi Akhilleus!

(have a square-root: √ )

Is the V value in the above equations simply the distance divided by time when the events are observed at rest? If so, solving for x' leads to a value of 0.

v is the speed of one observer relative to the other (the other way round, it's minus v of course).

Explain how you got a value of 0.

(btw, this problem really has nothing to do with relativity! )

 

[Akhilleus]

Using the distance between the two events divided by the time (5 microseconds) as V:

V = 1000m/(5 E-6)s = 2E8 m/s
g = 1/√(1 - (2E8)²/c²) = 1.8
x' = 1.8(1000m - (2E8)(5E-6)) = 0


So V is the speed of the moving observer as seen by a hypothetical second person at rest. Here, V is just distance between the two events divided by time. Would this be the same V required by the moving observer?

 

[tiny-tim]

So V is the speed of the moving observer as seen by a hypothetical second person at rest. Here, V is just distance between the two events divided by time. Would this be the same V required by the moving observer?

Yes, that's fine.

What's worrying you about that?

The question asks you to find a v such that x' = 0 …

that's what you've done! ​

(but why are you bothering with gamma? don't you see that you would have got the same result even if you knew no relativity?)

 

[Akhilleus]

Ah, I see. This was a LOT easier than I was making it... haha.

I was caught up on the idea that the distance traveled would contract if he was traveling at 0.6c, and thought that if he didn't slow down, event B would happen behind him. I'm very intrigued by the effects of relativity but I'm still not sure when they apply and when they don't.