Frames S and S\' are moving relative to each other along the x and x\' axes. The

Question

Frames S and S' are moving relative to each other along the x and x' axes. They set their clocks to t=t'=0 when their origins coincide. In frame S, event 1 occurs at x1=1 c*y and t1=1 y and event 2 occurs at x2=2.0 c*y and t2=0.5 y. These events occur simultaneously in frame S'. using spacetime diagrams (a) find the magnitude and direction of the velocity of S' relative to S? (b) at what time do both of these events occur as measured on S'?

this question is already posted but i'm having confusion on part b. How to I get a solid answer mathematically as opposed to just graphing it?

Explanation / Answer

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

The events occur simultaneously in frame S', therefore dt' is zero:

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

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

==> dt - (v/c^2) dx = 0

==> dt = (v/c^2) dx

==> (0.5y - 1y) = (v/c^2) (2 c.y - 1 c.y)

==> (-0.5y) = (v/c^2) (1 c.y)

==> (-0.5) = (v/c)

==> v = -0.5 c

t'1 = gamma (t1 - (v/c^2) x1)

t'1 = (1/(1 - (v/c)^2)^0.5) * (t1 - (v/c^2) x1)

t'1 = (1/(1 - (-0.5)^2)^0.5) * (1y - (-0.5/c)*(1c.y))

t'1 = (1/(1 - (-0.5)^2)^0.5) * (1.5 y)

t'1 = 1.7321 y

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