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Homework Statement
Spaceship is ##\scriptsize100m## long in its proper coordinate system and is moving relative to Earth. Astronaut sends a light signal from the rear part of the spaceship towards the front part. It seems to an observer on Earth, that a light signal needed ##\scriptsize 0.57\mu s##. What is the relative speed ##\scriptsize u## of the spaceship compared to Earth.
Homework Equations
- Lorentz transformations,
- time dilation,
- length contraction.
The Attempt at a Solution
I set spaceship's rear end in an origin of ##\scriptsize x'y'## and the observer on Earth in an origin of ##\scriptsize xy##. The problem states that length ##\scriptsize 100m## is in the proper coordinate system, so i know that ##\scriptsize \Delta x' = 100m \equiv \ell## (i ll denote proper length using ##\ell##). I also know that observer on Earth measured ##\scriptsize \Delta t = 0.56\mu s##.
Time events do not happen on the same place for neither of the observers (astronaut nor the observer on Earth), so i don't know if the time dilation eq. is reliable enough to calculate ##\scriptsize \gamma## and ##\scriptsize u ## afterwards. Can you confirm this please?
The speed of light is constant so:
\begin{align}
c=const.\quad
\left\{
\begin{aligned}
c&=\frac{\Delta x}{\Delta t} \xrightarrow{\text{i calculate $\Delta x$}} \Delta x = c \Delta t = 2.99\cdot10^{8}\tfrac{m}{s} \cdot 0.56 \cdot 10^{-6} s = 173.42m\\
c&=\frac{\Delta x'}{\Delta t'} \xrightarrow{\text{i calculate $\Delta t'$}} \Delta t' = \frac{\Delta x'}{c} = \frac{100m}{2.99\cdot 10^8 \tfrac{m}{s}} = 3.34\cdot10^{-7}m
\end{aligned}
\right.
\end{align}
1st it is weird to me that the length ##\scriptsize \Delta x > \Delta x'##. Especially because the problem states that ##\scriptsize \Delta x' \equiv \ell## which indicates that it must hold ##\scriptsize \Delta x' = \gamma \Delta x##. But it doesnt:
\begin{align}
\Delta x' &= \gamma \Delta x \xleftarrow{\text{because $\gamma \geq 1 \longrightarrow \Delta x'>\Delta x$}}\\
\gamma &= \frac{\Delta x'}{\Delta x}\\
\gamma &= \frac{100m}{173.42m}\\
\gamma &= 0.57
\end{align}
The ##\scriptsize \gamma## above is weird and i don't know how to continue to calculete the right ##\scriptsize u##. Please I need some explanation on this. Where did i do anything wrong? I think using the invariant interval ##\scriptsize \Delta x^2 - (c\Delta t)^2## could solve the problem fast, but please show me how to solve this using the Lorentz transformations or time dilation / length contraction equations.
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