# Extremely hard special relativity (2 particles with different speeds)

1. Jun 29, 2013

### 71GA

1. The problem statement, all variables and given/known data
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?

2. Relevant equations

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

3. The attempt at a solution
First i draw the image:
[Broken]

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). Lets 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 :)

Last edited by a moderator: May 6, 2017
2. Jun 29, 2013

### TSny

Check your numerical calculation of $\gamma_2$, I think you set it up ok but didn't evaluate it correctly.

In all of the equations that you wrote down, which quantity represents what you are asked to find?

3. Jun 29, 2013

### 71GA

Thank you. I was so sloppy. I did fix the $\gamma_2$. The right value is $\gamma_2 = 7.09$
First i need to find $\Delta t_2'$. By "all" did you mean A,B,C,D and E,F,G,H?

4. Jun 29, 2013

### TSny

Yes. So, which of equations E-H will get you the answer?

5. Jun 29, 2013

### 71GA

Equations (E), (F) and (H) i presume. Let me try this.

6. Jun 29, 2013

### Staff: Mentor

Here is what I think is a simpler way. Call the xy frame of reference S, and the frame of reference of the faster particle S'. Let both particles explode at time t = 0, as reckoned by the observers in the S frame of reference, and let the slower particle be located at x = 0, and the faster particle be located at x = 100 m at t = 0. As reckoned by the observers in the S' frame of reference, let the slower particle explode at x ' = 0 and at at time t' = 0 (according to the synchronized clocks of the S' frame of reference). So, for Event 1, you have:

x = 0, t = 0
x' = 0, t' = 0

Let Event 2 be the explosion of the faster particle. For event 2,

x = 100 m, t = 0
x' = ?? m, t' = ?? sec

You need to use the Lorentz Transformation to determine the ??s in S' for the second event.