How can the professor synchronize her students' clocks on a moving spaceship?

[fatfatfat]

Homework Statement

A physics professor on Earth gives an exam to her students who are on a spaceship traveling at speed v relative to Earth. The moment the ship passes the professor she signals the start of the exam. If she wishes her students to have time To (spaceship time) to complete the exam, show that she should wait a time (Earth Time) of

T = (To)sqrt[(1-v/c)/(1+v/c)]

before sending a light signal telling them to stop. (Hint: Remember that it takes some time for the second light signal to travel from the professor to the students.)

Homework Equations

Lorentz Transformations :
x'=γ(x-vt)
t'=γ(t-(vx/c^2))

Time dilation: t=γt'

t=d/v

The Attempt at a Solution

Well I first I defined all the different times.

To= total time in the spaceship frame

t= total time in Earth's frame = γTo = To/sqrt(1-(v^2/c^2))

tx= time it takes the professor's light signal to travel from her to the students = x/c

T= how long the professor should wait to send the second light signal after the first one = t-tx

x is the distance between the professor and the spaceship when the students receive the second signal.

I'm not really sure where to go from here. I know I've got to use x'=γ(x-vt) where x' is the distance that the spaceship has traveled before receiving the second signal somewhere, but I'm not sure what time to use in it and stuff like that.

Any help would be appreciated.

 

[robphy]

First, focus on the EVENTS.
(Can you draw a position-vs-time graph of what is happening [a spacetime diagram]?).
Then, it should be more clear what the various t's and x's mean.

 

[fatfatfat]

Umm..

Earth Frame:
at time t=0, the first signal is sent.
at time T the second signal is sent.
at time t the second signal is received by the spaceship. At this time both the spaceship and the light signal have traveled x.

Spaceship Frame:
at time t=0 the first signal is recieved.
at time To the second signal is recieved. The spaceship has traveled x'.

Am I not thinking of this correctly?

 

[fatfatfat]

Okay, so I've been playing around with stuff and this is what I've got:




Earth Frame:

tx=x/c because it's the amount of time it takes for the signal to reach the students.
so x= c(tx)

but x=vt also because t=x/v is the amount of time it takes the students to travel distance x.

So I said ctx=vt, therefore, tx=vt/c

then I filled it into

T= t-tx = t-vt/c = (ct-vt)/c = t(c-v)/c = t(1-v/c)

Since we already know that t=γTo I filled that in too and got

T= To(1-v/c) / sqrt[1-(v^2/c^2)]

which is starting to look a little bit like what I'm aiming for...




Does what I did make any sense or have I just totally lost track?

 

[fatfatfat]

I don't know if I'll get in trouble for this, but I just wanted to bring this back to the top before its pushed onto page 2 and forgotten... if that's not allowed just let me know, haha. Sorry.

 

[robphy]

Sorry, I've been busy today. I'll try and get back on this later. But maybe someone else will chime in.

 

[fatfatfat]

Figured it out! Thanks anyway :).

I was on the right track, all I did after that was square both sides of that last equation I had... and then it was clear from there!