Plane where clocks are synchronized in both frames

[LoadedAnvils]

Homework Statement

What is the velocity of the plane where clocks in two different frames are synchronized?

Homework Equations

Lorentz Transformations:

t' = γ(t - vx/(c^2))

x' = γ(x - vt)

Solution should be v(t = t') = (c^2)/v * (1 - 1/γ)

The Attempt at a Solution

I am getting the negative of the solution: v(t = t') = (c^2)/v * (1/γ - 1)

If I contract to the frame of the plane, I get that

γ_p(t - (v(t = t')/c^2)x) = γ_p(t' - (v(t = t')/c^2)x')

Simplifying and solving for v(t = t'), I get

v(t = t') = (c^2)(t - t')/(x -x')

Transforming t' to t and x, I get

v(t = t') = (c^2)(t - γt + γ(v/c^2)x)/(x - γx + γvt)

Dividing through by t on both numerator and denominator and noting that x/t = 0, I get

v(t = t') = (c^2)(1 - γ)/(γv)

Which leads to the solution I obtained.

EDIT: Never mind, I figured it out. I used the wrong form of the Lorentz transformation.

If anyone is interested, it's supposed to be t' = γ(t + vx/(c^2)) and x' = γ(x + vt)

 

[member38644]

instead of t' = γ(t - vx/(c^2)) and x' = γ(x - vt). This gives the correct solution of v(t = t') = (c^2)/v * (1 - 1/γ).