# Lorentz transformations question

• ElijahRockers
In summary, the conversation discusses finding the velocity of a pion in order to travel a distance of 10.0m. Using the Lorentz velocity transformation and the basic definition of velocity, the correct answer is determined to be .789c. There is also a discussion about the use of Lorentz transformations and the concept of Lorentz-invariant quantities in solving relativistic problems. The conversation ends with the suggestion to practice more problems to gain a better understanding of relativistic physics.

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## Homework Statement

The pion has an average lifetime of 26.0ns when at rest. For it to travel 10.0m, how fast must it move?

## Homework Equations

Lorentz velocity transformation?

## The Attempt at a Solution

I'm very lost... am I supposed to use u'x = (ux-v)/(1-vux/c2)? I thought I was following the lecture pretty well, but now that I'm trying to solve a problem, I have no idea what to do.

After some more poking around, I used Δt=γΔt'. Now I have one equation and two unknowns (v and Δt). I know I am supposed to use a velocity equation to find v now, but I'm still not sure which eqn to use, and how.

Last edited:
ElijahRockers said:
I'm very lost... am I supposed to use u'x = (ux-v)/(1-vux/c2)?
That's a velocity transformation. You won't need that.
After some more poking around, I used Δt=γΔt'. Now I have one equation and two unknowns (v and Δt). I know I am supposed to use a velocity equation to find v now, but I'm still not sure which eqn to use, and how.
How about using the most basic definition of velocity?

Ok, well I used v=d/t, plugging in my Δt for t, 10m for d, and I came up with the correct answer of .789c.

So I guess I still have a lot of thinking to do about when to use these transformations, I don't really get when exactly to use them, and I even get confused on when I need to use the inverse transformations or not, too.

I guess I have another question, assuming the moving frame is attached to the pion... then it observes itself to exist for 26E-9 seconds right? Then a stationary observer would perceive the pion to exist for over twice that amount of time, if I am thinking about this correctly.

So what about the distance, 10m, that we're talking about? Is that as observed by a stationary observer, or by the pion in its frame, or what?

Am I right in thinking that it's 10m as observed by a stationary observer since I used the 'stationarily observed' time duration in my velocity calcuation?

I feel like this stuff should come a lot more naturally to me, but for some reason I am really struggling with it...

ElijahRockers said:
Ok, well I used v=d/t, plugging in my Δt for t, 10m for d, and I came up with the correct answer of .789c.
Good.
So I guess I still have a lot of thinking to do about when to use these transformations, I don't really get when exactly to use them, and I even get confused on when I need to use the inverse transformations or not, too.
One way is to just list what you know, then consult your handy list of Lorentz transforms (See: Basic Equations of Special Relativity) to pick the one that works.

We are given Δx and Δt'. And you should know what Δx' is. (The events are the birth and death of the pion.) That tells you that you can use this transform:
$$\Delta x = \gamma(\Delta x' + v\Delta t')$$
I guess I have another question, assuming the moving frame is attached to the pion... then it observes itself to exist for 26E-9 seconds right?
Right. Let the rest frame of the pion be the primed frame.
Then a stationary observer would perceive the pion to exist for over twice that amount of time, if I am thinking about this correctly.
Well, you found the speed. Use it to calculate gamma. Then use Δt = γΔt'.
So what about the distance, 10m, that we're talking about? Is that as observed by a stationary observer, or by the pion in its frame, or what?

Am I right in thinking that it's 10m as observed by a stationary observer since I used the 'stationarily observed' time duration in my velocity calcuation?
Yes, the 10 m is the distance the pion is observed to move in the 'stationary' frame. How far does the pion move in its own frame? That's the same question I asked above: Δx' = ? (Which should be a trivial question, once things start to click.)
I feel like this stuff should come a lot more naturally to me, but for some reason I am really struggling with it...
Just about everyone struggles with this stuff. Just keep doing problems, thinking and rethinking the solutions. It will start to stick.

Doc Al said:
How far does the pion move in its own frame? That's the same question I asked above: Δx' = ? (Which should be a trivial question, once things start to click.)

I'm going to go out on a limb and say that Δx' = 0...

ElijahRockers said:
I'm going to go out on a limb and say that Δx' = 0...
Exactly. In the pion's rest frame, the pion doesn't move.

Doc Al has guided you in the right direction. As he says, it takes a bit of practice to get used to relativistic problems. Think about how many non-relativistic questions you have done in your lifetime. That's a lot of questions, right? So you've done comparatively only a small number of relativistic problems. (I know this is true for myself). So you should feel good that you are doing pretty well with relativistic physics, given that you have probably only a limited amount of practice with them. The more problems you do, the more familiar you will get. And also, I think that concepts such as lorentz-invariant quantities and the metric will become more useful when you do more problems, and you start to get a feel for how they are naturally useful ways of mathematically looking at relativistic problems.

Yeah I think I am getting the hang of it, maybe. :)
It's very interesting material, to say the least. Sometimes I wish I had majored in physics/astronomy, instead of engineering. :p

## 1. What are Lorentz transformations?

Lorentz transformations are a set of equations developed by Dutch physicist Hendrik Lorentz in the late 1800s. They describe how measurements of space and time are affected by the motion of an object or observer in relation to another object or observer. These equations are a key component of Albert Einstein's theory of special relativity.

## 2. Why are Lorentz transformations important?

Lorentz transformations are important because they allow us to understand how space and time behave in different reference frames, particularly at high speeds. They are essential for accurately describing phenomena such as time dilation and length contraction, which are crucial concepts in modern physics.

## 3. How do Lorentz transformations differ from Galilean transformations?

Lorentz transformations differ from Galilean transformations in that they take into account the constant speed of light and the relativity of simultaneity. Galilean transformations, on the other hand, assume that time and space are absolute and do not change based on an observer's reference frame.

## 4. What is the Lorentz factor?

The Lorentz factor, denoted by the symbol γ (gamma), is a term used in Lorentz transformations to calculate the effects of time dilation and length contraction. It is equal to 1 divided by the square root of 1 minus the square of the relative velocity between two objects, all multiplied by the speed of light, c.

## 5. How are Lorentz transformations applied in practical situations?

Lorentz transformations are applied in many practical situations, particularly in modern physics and engineering. They are used in the design of particle accelerators, GPS technology, and space travel. They also play a crucial role in understanding the behavior of subatomic particles and the effects of high-speed travel on human perception of time and space.