How Do Lorentz Transformations Affect Measurements of Time and Distance?

[Frillth]

Homework Statement

Events A and B are simultaneous in frame F and are 18 km apart on a line that defines the x-axis. A series of spaceships all pass at the same speed in the + x-direction, and they have synchronized their clocks so that together they make up a moving frame F'. They time events A and B to be separated by 0.80 microseconds. What is the speed of the spaceships? How far apart in space do they measure the two events to be?

Homework Equations

γ = 1/sqrt(1 - (u/c)^2)

(1) Δx' = γ(Δx - uΔt)
(2) Δt' = γ(Δt - uΔx)/(c^2)

The Attempt at a Solution

a. To get the speed, I used Δt = 0, Δt' = 8*10^-7 s, and Δx = 18000m. I plugged these into equation 2 and did a little bit of algebra to get:
Δt' = 4*10^6 m/s = c/75

b. To get the distance, I used Δx = 18000m, Δt = 0, and u = c/75, which I plugged into equation 1. When I did this, I got:
Δx' = 18001.6m

My answer to part a seems plausible, but my answer to part b just looks wrong to me. It seems like Δx' should not be so close to Δx. Are my solutions correct? If not, where did I mess up?

Thanks!

 

[alphysicist]

Homework Statement

Events A and B are simultaneous in frame F and are 18 km apart on a line that defines the x-axis. A series of spaceships all pass at the same speed in the + x-direction, and they have synchronized their clocks so that together they make up a moving frame F'. They time events A and B to be separated by 0.80 microseconds. What is the speed of the spaceships? How far apart in space do they measure the two events to be?

Homework Equations

γ = 1/sqrt(1 - (u/c)^2)

(1) Δx' = γ(Δx - uΔt)
(2) Δt' = γ(Δt - uΔx)/(c^2)

The Attempt at a Solution

a. To get the speed, I used Δt = 0, Δt' = 8*10^-7 s, and Δx = 18000m. I plugged these into equation 2 and did a little bit of algebra to get:
Δt' = 4*10^6 m/s = c/75

b. To get the distance, I used Δx = 18000m, Δt = 0, and u = c/75, which I plugged into equation 1. When I did this, I got:
Δx' = 18001.6m

I haven't checked all your calculations, but this number looks wrong to me. The length measured in frame F is the proper length, and length measurements in the other frames should be smaller.