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## Homework Statement

We have a 2 unstable particles moving in the ##\scriptsize xy## frame. The leading one has speed ##\scriptsize u_2=0.99c## while the other has speed ##\scriptsize u_1 < u_2##. When the distance between them is ##\scriptsize 100m## in the ##\scriptsize xy## frame they both seem to explode at the same time in ##\scriptsize xy##.

How much time does it pass between explosions in frame of the faster particle. Which particle does explode first in this system?

## Homework Equations

- Lorentz transformations (i think only for standard configuration)
- time dilation
- length contraction

## The Attempt at a Solution

First i draw the image:

First i calculate ##\scriptsize \gamma_2##:

\begin{align}

\gamma_2 = \frac{1}{\sqrt{1- 0.99^2}} = 7.09

\end{align}

I know is that

__if two events happen at the same time in some system that system measures proper length ##\scriptsize \ell##__. So i can conclude that ##\Delta x \equiv \ell##. With respect to this i can calculate the distance ##\scriptsize \Delta x_2' = \tfrac{1}{\gamma} \Delta x = \tfrac{1}{50.25}100m \approx 1.99m##.

**Question 1:**I

__think__that the calculated distance ##\scriptsize \Delta x'_2## is the distance between the events in frame ##\scriptsize x'_2,y'_2##

__if particle in system ##\scriptsize x'_1,y'_1## would be moving with speed ##\scriptsize u_2=u_1##__. I am not sure about my interpretation and need help.

Now i try to continue and use the fact that events happen at the same time in ##\scriptsize xy## frame ##\scriptsize \longrightarrow \Delta t = 0##. I know i have to use this in Lorentz transformations which i have to write

__2 times__(first time to connect the frame ##\scriptsize xy## with ##\scriptsize x'_1y'_1## and second time to connect frame ##\scriptsize xy## with ##\scriptsize x'_2y'_2## - i use ##\scriptsize \gamma_1, u_1## & ##\scriptsize \gamma_2,u_2## respectively). Let's do this:

Connecting the ##\scriptsize xy## with ##\scriptsize x'_1y'_1##:

\begin{align}

\Delta x &= \gamma_1 \left(\Delta x'_1 + u_1 \Delta t_1' \right) & \Delta t &= \gamma_1 \left(\Delta t_1' + \Delta x'_1 \tfrac{u_1}{c^2}\right)\\

\Delta x_1' &= \gamma_1 \left(\Delta x - u_1 \Delta t \right) & \Delta t_1' &= \gamma_1 \left(\Delta t - \Delta x \tfrac{u_1}{c^2}\right)\\

\end{align}

Connecting the ##\scriptsize xy## with ##\scriptsize x'_2y'_2##:

\begin{align}

\Delta x &= \gamma_2 \left(\Delta x'_2 + u_2 \Delta t_2' \right) & \Delta t &= \gamma_2 \left(\Delta t_2' + \Delta x'_2 \tfrac{u_2}{c^2}\right)\\

\Delta x_2' &= \gamma_2 \left(\Delta x - u_2 \Delta t \right) & \Delta t_2' &= \gamma_2 \left(\Delta t - \Delta x \tfrac{u_2}{c^2}\right)\\

\end{align}

If i apply the ##\boxed{\Delta t = 0}## i get some speciffic Lorentz transformations (which i marked A,B,C... for easier comunication):

Connecting the ##\scriptsize xy## with ##\scriptsize x'_1y'_1##:

\begin{align}

&(A) & \Delta x &= \gamma_1 \left(\Delta x'_1 + u_1 \Delta t_1' \right) & &(B) & 0 &= \gamma_1 \left(\Delta t_1' + \Delta x'_1 \tfrac{u_1}{c^2}\right)\\

&(C) & \Delta x_1' &= \gamma_1 \Delta x & &(D) & \Delta t_1' &= - \gamma_1 \Delta x \tfrac{u_1}{c^2}\\

\end{align}

Connecting the ##\scriptsize xy## with ##\scriptsize x'_2y'_2##:

\begin{align}

&(E) & \Delta x &= \gamma_2 \left(\Delta x'_2 + u_2 \Delta t_2' \right) & &(F) & 0 &= \gamma_2 \left(\Delta t_2' + \Delta x'_2 \tfrac{u_2}{c^2}\right)\\

&(G) & \Delta x_2' &= \gamma_2 \Delta x & &(H) & \Delta t_2' &= - \gamma_2 \Delta x \tfrac{u_2}{c^2}\\

\end{align}

**Question 2:**I noticed that (C) and (G) say that ##\scriptsize \Delta x'_1## and ##\scriptsize \Delta x'_2## will be larger than ##\scriptsize \Delta x##. I don't understand this yet and need some help here.

This is what i ve been able to do so far. I would appreciate if anyone would help me to finish i would be gratefull :)

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