Extremely hard special relativity (2 particles with different speeds)

[71GA]

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 :)

 

[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?

 

[71GA]

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

Thank you. I was so sloppy. I did fix the $\gamma_2$. The right value is $\gamma_2 = 7.09$

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

First i need to find $\Delta t_2'$. By "all" did you mean A,B,C,D and E,F,G,H?

 

[TSny]

I did fix the $\gamma_2$. The right value is $\gamma_2 = 7.09$

First i need to find $\Delta t_2'$.

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

 

[71GA]

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

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

 

[Chestermiller]

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.