Proving length contraction exists

[daselocution]

Homework Statement

Hey All; I am trying to follow along with the following example in my book, but I am getting lost somewhere.

"We can perform another gedanken experiment to arrive at the same result (showing that length contraction is described as L=L_o/γ; this time we lay the meterstick along the x' axis in the moving system K'. The two systems are aligned at t=t'=0. A mirror is placed at the end of the meterstick, and a flashbulb goes off at the origen at t=t'=0, sending a light pulse down the x' axis where it is reflected and returned. Mary (in the Moving system sees the stick at rest in the system K' and measures the proper length L_o (which should be 1 meter). Mary uses the same clock fixed at x'=0 for the time measurements. The stick is moving at speed v with respect to Frank (in the Fixed system K). The clocks at x=x'=0 both read zero when the origens are aligned just when the flashbulb goes off. Notice that in system K, by the time the light reaches the mirror the entire stick as moved a distance vt₁ and by the time the light has been reflected back to the front of the stick again the stick as moved another total distance vt₂; you should be able to find the solution in terms of length contraction as in the earlier examples.

I've tried it a few times and I'm really not sure where I keep on screwing up, but I cannot get derive the equation for length contraction using the lorentz transformations.

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

Go through your reasoning one step at a time. Spell it out for me - often when you explain things to someone else you get it.

Also see:
http://www.physicsguy.com/ftl/html/FTL_part1.html#sec:timedilation

 

[daselocution]

That response was at first frustrating but ultimately quite helpful. Does this make sense?:

When I started off doing the problem I was set on finding out what the length in Mary's system would be if it was transformed into Frank's system. Upon further thought, is the question instead asking how Mary would see what Frank is seeing? I know that in order to get the right answer I just need to flip my x=(x'+vt...etc.) into x'=(x-vt...) and the only reason I can come up with to say that flip makes sense is to say that we are actually trying to figure out how Mary would see what Frank is seeing, which is to say that we want the units as being in Mary's system, the one that starts with x', and to have that answer be based on the measurements that Frank makes in his own system of Mary's system. Does that follow? Am I thinking about it correctly then? This line of thought makes sense to me in that it takes into account the fact that Mary is moving with respect to Frank in the same way you could alternatively visualize Frank moving with -v with respect to Mary.

 

[Simon Bridge]

That response was at first frustrating but ultimately quite helpful. Does this make sense?

Well yes ... I've been doing this for a long time ;)

If I see how you are thinking, I can figure out how to explain things to you better - so I can work less but smarter - I'm not getting paid remember. ;) You will learn more, and retain what you have learned, and gain confidence in the subject.

Anyway - off your description:
Turning things upside down can be useful.

Mary has the "proper length", since she is stationary wrt the length being measured - she's just measuring it in a convoluted way.

You want to know what the length is in Frank's frame - in this F frame, the length is moving. Moving lengths are shorter than proper lengths. Try not to be distracted by the equations ... just use basic physics (v=d/t) inside each reference frame. The pattern should be the same as for the time-dilation one.

Take care - the flash-bulb goes off at the same place in both frames.

Start from
(1) $L+v\Delta t_1 = c\Delta t_1$
(2) $L-v\Delta t_2 = c\Delta t_2$
(3) $\Delta t = \Delta t_1 + \Delta t_2$

... take it a step at a time.

In Frank's frame, Mary's clock is slow: $\gamma \Delta t' = \Delta t$ (4)
(unprimed frame is Frank's.)

Note, however:

I cannot get derive the equation for length contraction using the lorentz transformations

... you realize that the equation for length contraction is (part of) the Lorentz transformation. The book is deriving it.

 

[daselocution]

Note, however: ... you realize that the equation for length contraction is (part of) the Lorentz transformation. The book is deriving it.

Ahh I did not actually realize that. Thank you very much, I was able to get it with that help.

 

[Simon Bridge]

That would help - yes.
It was one of those situations: was it a slip or was it really meant that the way it was written - better check JIC. Glad to be able to help :)