An introductory question about special relativity

[Yink]

Homework Statement

I can understand that the lightning coming from x=0 will reach to the assistant earlier, However, the question also asks for the time difference of the assistant receiving the lightning and this is the point I'm not sure about. My answer is 12km/c (c is the speed of light) Is this correct?

Relevant Equations

t=s/v

The way I was doing is that I list events
1. lightning hits x=0 this is (x_1=0,t_1)
2. lightning hits x=12 (x_2=12,t_2)
3. left lightning reaches "me" (x_3=9,t_3)
4. right lightning reaches "me" (x_4=9,t_4=t_3) t_4=t_3 since "I" see the lightning at the same time

Then the equation can be :
t_3=t_1+9/c
t_4=t_2+12-9/c=t_2+3/c

and since t_4=t_3 this implies t_2-t_1=6/c
Then it's possible to suppose t_1=0 and t_2=6/c

Then for the assistant:
1. lightning hits x=0 this is (x_1=0,t_1)
2. lightning hits x=12 (x_2=12,t_2)
3. left lightning reaches "me" (x_3'=3,t_3')
4. right lightning reaches "me" (x_4'=3,t_4'=t_3')

and t_3'=t_1+3/c=3/c
t_4'=t_2+12-3/c=6/c+9/c=15/c

The difference for the assistant is : 15/c-3/c=12/c

 

[TSny]

That all looks good. Your work is nicely and carefully done.

Once you work it out, you can sometimes step back and see other ways to get to the answer a little quicker.

For example, Let E be the event of the simultaneous arrival of the two flashes at x = 9 km.

It is clear that the flash from x = 0 must have passed x = 3 km at some time earlier than the occurrence of E. How much earlier? That's easy to write down in terms of $c$ and the distance between x = 3 km and x = 9 km.

Likewise, the flash from x = 12 km will pass x = 3 km at some time later than the occurrence of E. How much later is again easy to write down in terms of $c$ and the distance between x = 3 km and x = 9 km.

Then, the time difference between the arrival of the flashes at x = 3 km follows.

However, I like the way you explicitly wrote out the space and time coordinates (x, t) of the relevant events. When you get to more complicated problems where you are dealing with various events as observed in different frames of reference, you will find that writing out what you know about (x, t) for each event in each frame of reference will really help a lot.