Basic relativity problem - Lorentz Transformations

In summary, if an observer moves towards c at a speed of 2E8 m/s, they will observe the events at 500m and 1500m to be at the same point in space.
  • #1
Akhilleus
9
0

Homework Statement



Event A occurs at xA = 500m. Event B occurs 5 microseconds later at xB = 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.
 
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  • #2
Hi Akhilleus! :smile:

(have a square-root: √ :wink:)
Akhilleus said:
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. :confused:

(btw, this problem really has nothing to do with relativity! :biggrin:)
 
  • #3
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?
 
  • #4
Akhilleus said:
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. :smile:

What's worrying you about that? :confused:

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

that's what you've done! :smile:

(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?)
 
  • #5
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.
 

1. What is the Lorentz transformation?

The Lorentz transformation is a mathematical equation used to describe the relationship between space and time in the theory of special relativity. It was developed by the physicist Hendrik Lorentz and is a fundamental concept in understanding the effects of time dilation and length contraction at high speeds.

2. Why is the Lorentz transformation important?

The Lorentz transformation is important because it allows us to understand how time and space are perceived differently by observers moving at different speeds. This is essential to understanding the theory of special relativity and its implications for our understanding of the universe.

3. How does the Lorentz transformation work?

The Lorentz transformation is a set of equations that describe how time and space coordinates change when an observer moves at a constant velocity relative to another observer. It takes into account the speed of light and the principle of relativity to calculate these changes.

4. What is the difference between the Lorentz transformation and the Galilean transformation?

The Galilean transformation is an equation used to describe the relationship between space and time in classical physics, while the Lorentz transformation is used in the theory of special relativity. The key difference is that the Galilean transformation assumes that time and space are absolute, while the Lorentz transformation takes into account the effects of time dilation and length contraction at high speeds.

5. How is the Lorentz transformation applied in real-world situations?

The Lorentz transformation is applied in various fields, including physics, engineering, and space science. It is used to calculate the effects of relativity on spacecraft, GPS systems, and particle accelerators. It also plays a crucial role in understanding the behavior of particles at high speeds in particle physics experiments.

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