Help me with all of these please. Help me with all of these please. Help me with

Question

Help me with all of these please.

Help me with all of these please.

Help me with all of these please.

Help me with all of these please.

In the laboratory frame, event 1 occurs at x - 0 light-years, / = 0 years. Event 2 occurs at x = 6 light-years, / = 10 years. In all rocket frames, event 1 also occurs at the position 0 light-years and the time 0 years. The y- and z-coordinates of both events are zero in both frames. a In rocket frame A, event 2 occurs at time tf = 14 years. At what position xf will event 2 occur in this frame?

Explanation / Answer

The (proper time)^2 = t^2 - x^2 must be equal for all reference frames

in lab frame it is 10^2 - 6^2 = 64

A) this must be same in the frame of the rocket .Thus,

14^2 - x'^2 = 64

thus x' = 11.49 light years

B) the proper time must be same in all frames. Thus,

t''^2 - 5^2 = 64

tus t''= 9.43 light years

C) for event 2 to occur at the same point , x''' = 0, and propr time must be same in all frames. Thus,

we know delta(x') = (delta(x) - v*delta(t)) * gamma.    here gamma = 1/root(1-(v/c)^2)

we know delta x' = 0 , thus (delta(x) - v*delta(t)) * gamma = 0

=> delta(x) - v*delta(t) = 0

thus v= delta(x)/delta(t) = 0.6

thus velocity = 0.6c

D) the proper time must be same in all frames. Thus,

t'''^2 - 08^2 = 64

thus t''' = 8 years.

here in all forms we have assumed c=1 and v= fraction of the speed of light.

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