Relativity Question with Lorentz tranformations

[sr6622]

Homework Statement

Observer O sees a red flash of light at the origin at t=0 and a blue flash of light at x = 3.26km at a time t = 7.63 $$\mu$$ seconds. What are the distance and the time interval between the flasheds according to ovserver O', who moes relative to O in the direction of increasing x with a speed of 0.625c? Assume that the origins of the two coordinate systems line up at t = t' = 0.

Homework Equations

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

t' = $$\gamma$$t - (u/c^2)x

The Attempt at a Solution

I solved for t' and got the answer 1.73 $$\mu$$seconds. From the back of my textbook, this seems to be the correct answer.

When I put x = 3.26km into the first equation above, I always get the answer to be 2.34 km. The answer fromt the back of my textbook says $$\Delta$$ x' = -1.90 km.

I'm wondering what I'm doing wrong...

 

[dynamicsolo]

Just to make it a little clearer, this equation should be written

t' = $$\gamma$$[t - (u/c^2)x]

I solved for t' and got the answer 1.73 $$\mu$$seconds. From the back of my textbook, this seems to be the correct answer.

When I put x = 3.26km into the first equation above, I always get the answer to be 2.34 km. The answer fromt the back of my textbook says $$\Delta$$ x' = -1.90 km.

I'm wondering what I'm doing wrong...

Are you sure you have the answer for the right problem? (I'm hoping they didn't change the numbers in the problem between editions and forget to change the answers -- I've seen that a little too often...)

Something about their answer doesn't seem right because the spacetime interval should be

(delta-s)^2 = (c delta-t)^2 - (delta-x)^2 = (c delta-t')^2 - (delta-x')^2 ,

which doesn't happen with their values.

I have $$\gamma$$ = 1.2810 . I guess we're agreed that x' = 0 , t' = 0 for the red flash. I also agree with you that, for the blue flash,

x' = (1.281) [ (3.26 km) - (0.625)(300,000 km/sec)(7.63e^-6 sec) ] = 2.343 km .

However, I get

t' = (1.281) [ (7.63e^-6) - (0.625)(3.26 km/300,000 km/sec) ]

= 1.074e^-6 sec or

ct' = 0.322 km.

In O's frame,

(delta-s)^ 2 = [ (300,000 km/sec)(7.63e^-6 - 0 sec) ]^2 - ( 3.26 - 0 km )^2
= -5.388 km^2

A negative space-time separation makes sense because the light-travel time in O's frame from the location of the blue flash is 3.26 km/300,000 km/sec = 10.9 microseconds, which is longer than the time separation between the flashes.

In O' 's frame,

(delta-s)^ 2 = [ (300,000 km/sec)(1.074e^-6 - 0 sec) ]^2 - ( 2.343 - 0 km )^2
= -5.386 km^2 (the difference is likely "round-off" error) .

You do not get this value with the book's answers. Unless we're missing some point about the problem description, I find their answer suspect.