- #1
Tubefox
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Homework Statement
A traveler in a rocket of length 2d sets up a coordinate system S' with origin O' anchored at the exact middle of the rocket and the x' axis along the rocket’s length. At t' = 0 she ignites a flashbulb at O'. (a) Write down the coordinates t'_F, x'_F, and t'_B, x'_B for the arrival of the light at the front and back of the rocket. (b) Now consider the same experiment as observed in a frame S relative to which the rocket is traveling at speed v (with S and S' arranged in the standard configuration). Use the Lorentz transformation to find the coordinates xF , t_F and xB , tB of the arrival of the two signals.
(This is the part my question is on) Repeat but do not use Lorentz transformation, just use length contraction and the fact that the speed of light is the same in every reference frame. Follow these steps:
1) Sketch the contracted rocket in the S frame at t=0. It moves with speed v.
(I did this.)
2) Write a formula for the time tF when the front of the rocket gets to xF
3) Write a formula for the time tF when the light gets to xF
4) Equate (2) and (3) to solve for xF and tF
5) Repeat for xB and tB
Homework Equations
[itex]L = \frac{L_0}{\gamma}[/itex]
[itex]x=\gamma(x' + vt')[/itex]
[itex]t=\gamma(t' + \frac{vx'}{c^2})[/itex]
The Attempt at a Solution
Here's the prime frame measurements:
[itex]x'_F=d \\
t'_F = \frac{d}{c}\\
x'_B = -d\\
t'_B= \frac{d}{c}[/itex]
And according to the Lorentz transform equations, here's what we have for S:
[itex]x_F=\gamma d (1+\frac{v}{c})\\
t_F=\frac{\gamma d}{c}(1+\frac{v}{c})\\
x_B=\gamma d(\frac{v}{c}-1)\\
t_B=\frac{\gamma d}{c}(1-\frac{v}{c})[/itex]
That's all fine. The issue arises when I try to follow the procedure outlined in the problem statement. It appears to be nonsense. Why would it take the same amount of time for the rocket to get to [itex]x_F[/itex] as it would for the light to get to [itex]x_F[/itex]? Even so, how can I solve for two unknowns based on one equation? Could someone, at the very least, clarify the wording a little bit?
Thanks.