Special Relativity with Three reference frames

[scienceLilly75]

Homework Statement

In the laboratory frame, event 1 occurs at x = 0 light-years, t = 0 years. Event 2 occurs at x = 6 light-years, t = 10 years. In all rocket frames, event 1 also occurs at the position 0 light-years and the time 0 years. The y- and z- coordinates of both events are zero in all frames.
a) In rocket frame A, event 2 occurs at time t’ = 14 years. At what position x’ will event 2
occur in this frame?
b) In rocket frame B, event 2 occurs at position x’’ = 5 light-years. At what time t’’ will
event 2 occur in this frame?
c) How fast must rocket frame C move if events 1 and 2 occur at the same place in this
rocket frame?

Homework Equations

s^2 = c^2Δt^2-Δx^2
t = (1/√1-v^2/c^2)t'

The Attempt at a Solution

For part A, I did c^2Δt^2-Δx^2 = c^2Δt'^2-Δx'^2. Because the everything in the rocket frames start at 0, then the entire left side becomes 0 and I get. √c^2 * (14)^2 = Δx' which is 4.2*10^9 or 14c.
I did the same thing for part B. The c^2Δt^2-Δx^2 = c^2Δt''^2-Δx''^2 and ended up with √5/c = 1.29*10^-4 = Δt''.
For part c, I know that I need to use t = (1/√1-v^2/c^2)t' . However, if I set t=0, then I get v rocket = c ---which shouldn't be right. It should be some fraction of c. My question is if if I am looking at a third reference frame, then do I use the values for time and position from the one before it? So to find the v in rocket frame C, I need to only use the data from rocket frame B?

 

[tiny-tim]

Welcome to PF!

Hi scienceLilly75! Welcome to PF!

(try using the X² button just above the Reply box )

In the laboratory frame, event 1 occurs at x = 0 light-years, t = 0 years. Event 2 occurs at x = 6 light-years, t = 10 years. In all rocket frames, event 1 also occurs at the position 0 light-years and the time 0 years. The y- and z- coordinates of both events are zero in all frames.
a) In rocket frame A, event 2 occurs at time t’ = 14 years. At what position x’ will event 2
occur in this frame?
…
For part A, I did c^2Δt^2-Δx^2 = c^2Δt'^2-Δx'^2. Because the everything in the rocket frames start at 0, then the entire left side becomes 0 and I get. √c^2 * (14)^2 = Δx' which is 4.2*10^9 or 14c.

No …

i] if x is in light-years, and t is in years, then c = … ?

ii] the separation (c²t² - x²) has to be between events 1 and 2, so it isn't 0 !

 

[scienceLilly75]

But at the beginning the time and position of the event is 0, so the left side has to be zero right?

Or do I need to find spatial separation between A and B, then spatial separation between B and C?

 

[tiny-tim]

Hi scienceLilly75!

(just got up :zzz:)

But at the beginning the time and position of the event is 0, so the left side has to be zero right?

Or do I need to find spatial separation between A and B, then spatial separation between B and C?

sorry, but that doesn't even make sense …

A B and C are frames, and there's no separation between frames (spatial or otherwise)

separation is between two events …

so which two events are you going to use? ​

(oh, and c = … ? )

 

[scienceLilly75]

I am confused. I understand that there is no separation between frames but between events in frames. In that case, I am stuck. The only thing I know about the frame C is that event one occurs at t=0, x=0. If the events occur on the same place on the rocket, then x' also equals 0. But then that will leave me with t' = 0 using the spatial relativity equation. if t'=0, then it is not moving. Where am I going wrong?

 

[tiny-tim]

Hi scienceLilly75!

I understand that there is no separation between frames but between events in frames.

Why do you say "in frames"? Separation is between two events, period.

Use event 1 and event 2. ​

 

[scienceLilly75]

So to find the speed, I have to use info from event 1 and 2 how they appear in frame c? then I can use v = x/t?

 

[Chestermiller]

Make up a table. Label the rows Laboratory, Rocket A, Rocket B, Rocket C. Label the columns Event 1 Time, Event 1 Distance, Event 2 Time, Event 2 distance. Fill in the table with any information you have available. If there is information missing, fill in any algebraic variable name that you choose. Add a 5th column to the table in which you calculate the following quantity

((Event 2 Time) - (Event 1 Time))² - ((Event 2 Distance) - (Event 1 Distance))². This quantity must be the same for all rows of the table.

 

[tiny-tim]

So to find the speed, I have to use info from event 1 and 2 how they appear in frame c? then I can use v = x/t?

why do you need for speed for part (a) ?

what is the separation between events 1 and 2 ?​

 

[Chestermiller]

Question 3 is a little ambiguous. I guess it is asking how fast rocket C is traveling relative to the laboratory frame of reference. The key to answering this question is recognizing that rocket C is physically present at both events.