24.8 Lorentz Transformation: Examples

Imagine a team of observers (Team Red) are riding a red broomstick, and zip by Earth at speed $v$. Assume the origins of Team Earth and Team Red cross at $t = t^\prime=0$ (that is, $(t=0, x=0) = (t^\prime= 0, x^\prime= 0)$). Imagine that a member of Team Red located at their origin has stolen a top secret file. So, you jump on your new super-broomstick, and chase after Team Red at speed $V$ which is larger than $v$. You leave Earth at time $t = t_1$. (To distinguish the various $\gamma$ factors, write $\gamma[w]= \frac{1}{\sqrt{1-w^2}} $ for any $w$ expressed in units with $c = 1$).

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1. Question
At what time $t_2$ according to Earth clocks do you catch up with the outlaw in Team Red?
- $\frac{Vt_1}{V - v}$
- $\frac{vt_1}{V - v}$
- $ \frac{(v+ V) t_1}{V}$
- $\frac{(v+ V) t_1}{V - v}$
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Question 2 of 5

2. Question
What is the time according to Team Red when you start chasing after them?
- $t_1$
- $t_1/\gamma [v]$
- $t_1/\gamma [V]$
- $t_1\gamma [v]$
- $t_1\gamma [V]$
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Question 3 of 5

3. Question
According to Team Red, how far away are you from the outlaw when you start chasing them?
- $vt_1/\gamma [v]$
- $vt_1\gamma [v]$
- $vt_1/\gamma [V]$
- $vt_1\gamma [V]$
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Question 4 of 5

4. Question
According to Team Red, at what time do you catch up with the outlaw?
- $\frac{Vt_1}{V - v}$
- $\frac{vt_1}{V - v}$
- $\frac{Vt_1}{(V - v)\gamma[v]}$
- $\gamma[v] \frac{(v+ V) t_1}{V - v}$
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Question 5 of 5

5. Question
According to your watch (which read $t_1$ when you left Earth), what time is it when you catch up with the outlaw? (Don't concern yourself with the effects of acceleration. Instead, if you are bothered, imagine that the question is more precise: Another observer moving past Earth with speed $V$, passes the origin of the Earth's frame at time $t_1$ on their own clock (and on Team Earth's clock). What time will that observer say it is when they catch up with the outlaw who is at the origin of Team Red?)
- $t_1(1 + \frac{v}{(V-v)})$
- $t_1(1 + \frac{v\gamma[V]}{(V-v)\gamma[v]})$
- $t_1(1 + \frac{v}{(V-v)\gamma[v]})$
- $t_1(1 + \frac{v}{(V-v)\gamma[V]})$
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Module 24: Lorentz Transformation: Examples - Problem 3

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