Understanding Lorentz Transforms: Interpreting Results and Visualizing Answers

[strangequark]

Ok, this is my last study problem, I think I got it, but my answers seem a little odd...

Homework Statement

A rocket of length $$1000 meters$$ is at rest in S'. The nose of the rocket is at$$x'=0$$and the tail of the rocket is at$$x'=-1000 meters$$. S' is moving with a velocity of $$v=\frac{3c}{5}$$ in the positive x direction relative to S.

Four events are given:

Event A is the synchronizing event where the nose of the rocket is at the origin in both frames:
$$x_{A}=x'_{A}=t_{A}=t'_{A}=0$$

Event B is simultaneous with A in S:
$$t_{B}=t_{A}=0$$

Event C is when the tail of the rocket passes the origin as observed in S

Event D is simultaneous with C and is when an observer in S sees the nose of the rocket pass by him.

Homework Equations

$$x'=\gamma(x-vt)$$

$$t'=\gamma(t-\frac{vx}{c^{2}}$$

I also used,

$$L=\frac{L_{0}}{\gamma}$$

The Attempt at a Solution

This is why I'm worried... it seems straight-forward...

All I did was calculate the length of the rocket as observed in S:

$$L=\frac{1000 meters}{5/4}=800 meters$$

I believe this gives me spatial coordinates for all of the events...

$$x_{C}=x_{A}=0$$

$$x_{B}=-800 meters$$

$$x_{D}=800 meters$$

as well as temporal coordinates:

$$t_{A}=t_{B}=0$$ (A is given in the problem and A,B are simultaneous)

$$t_{C}=t_{D}=\frac{x_{C}-x_{B}}{v}=\frac{x_{D}-x_{A}}{v}=4.4475 x 10^{-6} seconds$$

Then I just applied the coordinate transforms, and got:

$$x'_{A}=0$$ (given)
$$x'_{B}=-1000 meters$$
$$x'_{C}=-1000 meters$$
$$x'_{D}=0$$

$$t'_{A}=0$$
$$t'_{B}=2.0014 x 10^{-6} seconds$$
$$t'_{C}=5.5559 x 10^{-6} seconds$$
$$t'_{D}=3.55802 x 10^{-6} seconds$$

Now, everything here looks a little weird... events B and C in the S' frame happen in the same place? And the sequence of events in S' is A-B-D-C?

Am I mis-applying the transforms or misinterpreting the problem?

If not, could someone please help me with interpreting the answers?

Much thanks in advance!

 

[learningphysics]

What is the question? calculate when all 4 events happen in both frames?

 

[strangequark]

yes, sorry, it's basically fill in all the unknown times...

 

[learningphysics]

yes, sorry, it's basically fill in all the unknown times...

What is event B?

 

[learningphysics]

I'm also confused about events C and D... if within the same frame, they happen at the same position at the same time... then they represent the same event. is the observer described in D at the origin in S?

 

[strangequark]

Sorry, my description was bad...

Event B is when an observer in S (not at the origin) sees the tail of the rocket pass over his head at the same time that an observer at the origin in S sees the nose of the rocket pass over HIS head...

For events C and D, they happen at the same time, but not the same position... the observer of event C is at the origin, while the observer of D is not (both observers are in the S frame though).

 

[learningphysics]

Your answers all look right to me. Only thing is the decimal places I think...

tc is exactly 40/9 * 10^-6 = 4.444 * 10^-6

tb' = 2.00*10^-6 exactly
tc' = (50/9)*10^-6 = 5.556 * 10^-6
td' = (32/9)*10^-6 = 3.556 * 10^-6

 

[strangequark]

Great, thanks again... I feel like I have a decent understanding of how to set up these problems as I do them, but everytime I come to an answer that I can't visualize it makes me wonder... I suppose I'm used to being able to tell if an answer is reasonable just by looking at the problem and I can't seem to do that yet with the Lorentz transforms... I'll keep at it, eventually it will click.