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.

 

[baba89]

Your solutions for part a and b are correct. The Lorentz transformations are used to calculate how space and time measurements change between two frames of reference that are in relative motion. In this case, the spaceships are moving at a high speed compared to frame F, which causes a significant difference in the measurement of space and time between the two frames. This is why the distance between events A and B, measured in frame F', is slightly longer than the distance measured in frame F. It may seem counterintuitive, but this is a fundamental concept in special relativity. Your answer for the speed of the spaceships is also correct, and it is important to note that it is relative to frame F, not frame F'. Great job on your solution!