Lorentz transformations question

[ElijahRockers]

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 = (u_x-v)/(1-vu_x/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.

 

[Doc Al]

I'm very lost... am I supposed to use u'_x = (u_x-v)/(1-vu_x/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?

 

[ElijahRockers]

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...

 

[Doc Al]

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.

 

[ElijahRockers]

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...

 

[Doc Al]

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.

 

[BruceW]

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.

 

[ElijahRockers]

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