Please Help me out with all three part! Thanks!!! LS1 events in two different fr

Question

Please Help me out with all three part! Thanks!!!

LS1 events in two different frames In a stationary frame S one has event A taking place at the origin, a(0,0,0,0); and a second event B with coordinates B(10c,0,0,8) where the numbers 10, 8 are in seconds. A second frame S' is moving in the x-direction and is coincident with frame S at the origin at t = t' = 0. What must be the velocity of the moving frame for the two events A, and B to take place simultaneous in the moving frame? With the velocity determined in a, what is the spatial separation of the two events in the moving frame? Use the Lorentz transformations to solve parts a, b. Verify that the spacetime interval for these two events in frames S and S' are the same.

Explanation / Answer

a)

dt' = gamma (dt - (v/c^2) dx)

dt' = 0

gamma = 1/(1 - (v/c)^2)^0.5

==> 0 = 1/(1 - (v/c)^2)^0.5 (8 - (v/c^2) * 10c)

==> 0 = 1/(1 - (v/c)^2)^0.5 (8 - (v/c) * 10)

==> v = 0.8 c = 2.4x10^8 m/s

b)

dx' = gamma (dx - v dt)

==> dx' = 1/(1 - (v/c)^2)^0.5 (10*3e8 - 2.4e8*8)

==> dx' = 1/(1 - (0.8)y2)y0.5 * (10*3e8 - 2.4e8*8)

==> dx' = 18x10^8 meters = 6c

c)

s^2 = dx^2 + dy^2 + dz^2 - c^2 dt^2 = (10c)^2 + 0 + 0 - c^2 * 8^2 = 36 c^2

s'^2 = dx'^2 + dy'^2 + dz'^2 - c^2 dt'^2 = (6c)^2 + 0 + 0 - 0^2 = 36 c^2 = 36 c^2

====> spacetime intervals are the same.

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