Lorentz Transformation For Moving Particles

In summary, the Lorentz transformation describes how the measurements of time and space change for observers in different inertial frames, particularly when dealing with moving particles. It accounts for the effects of relative velocity close to the speed of light, demonstrating that time can dilate and lengths can contract depending on the observer's motion. This transformation is essential for understanding phenomena in special relativity, ensuring that the laws of physics remain consistent across different frames of reference.
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Homework Statement
Consider the experiment of Problem 1.26 (shown below) from the point of view of the pions' rest frame. In part (c) how far (as "seen" by the pions) does the laboratory move, and how long does this take? How many pions remain at the end of this time?
Relevant Equations
## x' = \gamma (x - vt) ## or ## x = \gamma (x' + vt') ##
So, I linked an image to Problem 1.26 below. As far as that problem, to save you the trouble, the answers (at least from what I have) are:
a) ##\gamma = 1.7 ##
b) ## t = 3x10^{-8} ##
c) ## Pions = 30,555 ##
d) ## Pions = 29,628 ##

Fairly confident those are correct, but now this question posted above is throwing me off.

For the first part, how far does the laboratory move, I believe the equation to use would be ## x' = \gamma (x - vt) ## or ## x = \gamma (x' + vt') ##, but I'm not sure exactly what would get used, and what values. And the problem is that I tried using both, and both give me values much higher than the 36m that the lab observes the pions to move. I don't feel this is right, but maybe I'm looking at it all wrong?

And as far as how long it takes, again I am confused, because I feel that's the same value I already had to calculate for the previous problem, to find out the length of time they experience for the distance traveled (to calculate how many would be left after the 36m distance). That would be ## t = 1.2x10^{-9}s ##, compared to the ## 2x10^{-9}s ## the lab observes. This makes sense, since the pions would experience less time passed within that timeframe, hence why there are more particles than expected based solely on the half-life. So, are they simply asking for the answer I already got of the ## 1.2x10^{-9}s ## from question c in 1.26?

And as far as how many remain, this is even more confusing to me. They are saying to consider the experiment from 1.26, and part c of that already asked how many pions would be left at the end of that time, and now, they seem to be asking the same exact question again, only from the perspective of the pions??? It's still the same exact event within the same time frame, so I would think it would be the pions would "see" the same exact amount of pions remaining within that distance traveled as the lab had. I don't see how, if they are asking after they traveled 36m by the perspective of the lab, how many are left, how it could possibly be a different value than the one from the lab?

I'd appreciate any help anyone can provide, because I felt like I was getting this in 1.26, but now with this problem, I feel like it has thrown me off.


Problem 1.26.JPG
 
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As reckoned from the laboratory frame, the pions are traveling at 0.8c and, as reckoned from the pion frame, the laboratory is traveling 0.8c. So, for the pions to travel 36 m as reckoned from the laboratory frame, the two relevant events are

t = 0, x = 0, t' = 0 , x ' = 0

t = 36/(0.8c) and x = 36, x'=0, t'=?
 
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FAQ: Lorentz Transformation For Moving Particles

What is the Lorentz Transformation?

The Lorentz Transformation is a set of equations in physics that relate the space and time coordinates of two observers moving at constant velocity relative to each other. It is a fundamental aspect of Einstein's theory of special relativity, allowing for the calculation of how measurements of time and space change for observers in different inertial frames.

How does the Lorentz Transformation affect measurements of time and distance?

The Lorentz Transformation shows that time and distance are not absolute but depend on the relative motion of the observer. For example, time dilation occurs when a moving clock is observed to tick more slowly compared to a stationary clock, while length contraction means that an object in motion is measured to be shorter along the direction of motion from the perspective of a stationary observer.

What are the key equations of the Lorentz Transformation?

The key equations of the Lorentz Transformation are: 1. \( x' = \gamma (x - vt) \) 2. \( t' = \gamma \left( t - \frac{vx}{c^2} \right) \) where \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \) is the Lorentz factor, \( v \) is the relative velocity, \( c \) is the speed of light, and the primed coordinates represent the moving observer's frame.

What is the significance of the Lorentz factor?

The Lorentz factor (\( \gamma \)) is significant because it quantifies the amount of time dilation and length contraction experienced by an object moving at a relativistic speed. As the velocity \( v \) approaches the speed of light \( c \), \( \gamma \) increases dramatically, leading to significant relativistic effects that cannot be ignored in high-speed scenarios.

Can the Lorentz Transformation be applied to all types of motion?

The Lorentz Transformation is specifically applicable to inertial frames of reference, which are frames that move at constant velocity. It does not apply to non-inertial frames, where acceleration is involved. In such cases, the principles of general relativity may need to be invoked to account for the effects of gravity and acceleration on time and space measurements.

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