Far, far away?

Classical Mechanics Level 2

If you have not done the previous set , I recommend you do it first.

Once we have the concept of synchronized clocks and a definition of how to relate the time coordinates of observers at different points we can define a distance. Let's choose a region of space with a family of observers moving through it, all with clocks synchronized by sending light pulses back and forth. Before when dealing with time we used the variable $F$ for Finn's time and $A$ for Aaron's time. Since we now have synchronized clocks, we can define a variable $t$ as the coordinate time which is a valid, well defined coordinate everywhere in the region of interest.

Consider the scenario where Finn and Aaron are communicating with light pulses. At time $t$ $_1$ , Finn sends a light pulse to Arron, who reflects is back, and is received by Finn at time $t$ $_2$ . Which of these is a possible definition for the distance $x$ between Finn and Aaron?

x=t_2

x=(t_2+t_1)/2

x=t_1

x=(t_2-t_1)/2

1 solution

Siam Habib
Jun 25, 2014

First of all notice that $x = t_1$ is not true. Because on that case, if Finn sent a light pulse $5$ minutes later, the distance would increase.

Similarly, $x = t_2$ can't also be true. Because on that case , if Finn had sent a light pulse $5$ minutes earlier, the light pulse would return $5$ minutes earlier and the distance would decrease.

Same goes for, $x = \frac{t_1+t_2}{2}$ .[ If Finn had sent it $x$ minutes earlier or later the distance would decreased or increased $x$ ]

Now, let's examine the option $x = \frac{t_1-t_2}{2}$ . On this case distance is independent to when the event occurs. So, this is our answer.

Also, notice that $distance = velocity\Delta time$ . On this case $velocity$ is just considered as $1$ . So, this indeed our answer.

Far, far away?

1 solution

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