Relativity and Lorentz Transformations

In summary, the conversation discusses a high-energy accelerator experiment with two particles approaching each other head-on at a speed of 0.9500c in the laboratory frame of reference. The question asks for the magnitude of one particle's velocity relative to the other. The solution involves using the equation v_x' = (v_x - u) / (1 - (uv_x/c^2)) in a new frame of reference that is moving with velocity u. In this new frame, the velocity of the particle is v_x'. The conversation also mentions a similar problem involving two protons moving away from each other and asks what an observer in the rest frame of the Earth would measure for the speed of each proton. The same equation can be used
  • #1
mhz
39
0

Homework Statement



Two particles in a high-energy accelerator experiment are approaching each other head-on, each with a speed of 0.9500c as measured in the laboratory.

What is the magnitude of the velocity of one particle relative to the other?

Homework Equations



[itex]v_x^'\frac{v_x-u}{1-\frac{uv_x}{c^2}}[/itex]


The Attempt at a Solution



I've considered the laboratory to be moving in addition to one particle having twice the given speed and the other zero, but I don't know what I'm actually doing. I really would look a nice, concise explanation of what is going on. Thank you.

Additionally, there is a similar problem that I am completely lost with:

Two protons are moving away from each other. In the frame of each proton, the other proton has a speed of 0.615c.

In the rest frame of the Earth the protons are moving in the opposite directions with equal values of speed. What does an observer in the rest frame of the Earth measure for the speed of each proton?

If you are interested in previous responses to a similar question, see here: https://www.physicsforums.com/showthread.php?t=481467
 
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  • #2
A particle travels with velocity vx along the x-axis in the laboratory frame of reference. In an other frame of reference that travels with velocity u with respect to the laboratory, the velocity of the particle is vx'. If you are an observer, sitting in that new fame of reference, you would see the particle traveling with vx'.

[tex]v_x'=\frac{v_x-u}{1-\frac{u v_x}{c^2}}[/tex].

Substitute the velocity of one particle for u and the velocity of the other one for v.

ehild
 

Related to Relativity and Lorentz Transformations

1. What is the theory of relativity?

The theory of relativity is a scientific theory developed by Albert Einstein in the early 20th century. It is a theory that explains the relationship between space and time, and how they are affected by gravity. There are two main components of the theory: the special theory of relativity and the general theory of relativity. The special theory deals with objects moving at constant speeds, while the general theory deals with objects under the influence of gravity.

2. What are Lorentz transformations?

Lorentz transformations are a set of equations that describe how space and time coordinates change for an observer moving at a constant velocity relative to another observer. They were first derived by Hendrik Lorentz in the late 19th century and were later incorporated into Einstein's theory of special relativity.

3. How do Lorentz transformations relate to the theory of relativity?

Lorentz transformations are a key component of the theory of relativity. They help to explain how time and space are relative to the observer's frame of reference and how they can change depending on the observer's relative motion. Without Lorentz transformations, the theory of relativity would not be able to accurately describe the effects of motion on space and time.

4. Can you give an example of how Lorentz transformations work?

One example of Lorentz transformations in action is the phenomenon of time dilation. This occurs when an object is moving at high speeds relative to an observer, causing time to appear to pass slower for that object. This is due to the time component of the Lorentz transformation, which shows that time is not constant for all observers and can be affected by motion.

5. How have Lorentz transformations been tested and proven?

Lorentz transformations have been extensively tested and have been shown to accurately predict the behavior of objects at high speeds. One famous example is the measurement of the muon lifetime, which has been observed to increase due to time dilation in accordance with Lorentz transformations. Additionally, the GPS system relies on the principles of relativity and Lorentz transformations to accurately calculate locations on Earth.

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