1) Given x' = 1/a (x - vt) and t' = 1/a (t - vx/c

^{2}),

Then: x' = x/a - vt/a and t' = t/a - vx/ac

^{2}

and: x' + vt/a = x/a and t' + vx/ ac

^{2 }= t/a

so:

x = a(x' + vt/a) and t' = a(t' + vx/ ac

^{2 })

finally: x = a(x' + vt) and t = a(t' + vx/c

^{2})

2) We have: x' = 60m, t' = 8 x 10

^{-8}s and y' = y, z' = z

v = 0.6c = 1.8 x 10

^{8}m/s

Then:

x = [60m + (1.8 x 10

^{8}m/s)( 8 x 10

^{-8}s)]/ (0.64)

^{½}

x = [60m + 14.4m]/ 0.8 = 74.4m/0.8 = 93m

and t =

[(8
x 10

t = 2.5 x 10

The space time coordinates are: (93 m, 2.33 x 10

3) The problem requires no relative motion defined specifically in the x-direction so the equations:

t = t' + x'v/c

and

t' = t - xv/c

are immediately simplified by the terms in x becoming zero, so:

t = t'/(1 - v

and

t' = t /(1 - v

Here: t = time passage on Earth clock

and t' = time passage on astronaut's clock

For t = 1 Earth year = 365 ¼ days:

t' = (365 ¼ days)/ [1 - (0.9c)

^{-8}s)+ (1.8 x 10^{8}ms^{-1})(60m)/(3 x 10^{8}ms^{-1})^{2}/0.8t = 2.5 x 10

^{-7}s/ 0.8 = 2.33 x 10^{-7}sThe space time coordinates are: (93 m, 2.33 x 10

^{-7}s)3) The problem requires no relative motion defined specifically in the x-direction so the equations:

t = t' + x'v/c

^{2}/(1 - v^{2}/c^{2})^{½}and

t' = t - xv/c

^{2}/(1 - v^{2}/c^{2})^{½}are immediately simplified by the terms in x becoming zero, so:

t = t'/(1 - v

^{2}/c^{2})^{½}and

t' = t /(1 - v

^{2}/c^{2})^{½}Here: t = time passage on Earth clock

and t' = time passage on astronaut's clock

For t = 1 Earth year = 365 ¼ days:

t' = (365 ¼ days)/ [1 - (0.9c)

^{2}/c^{2}]^{½}
=
(365 ¼ days)/(1 - 0.81)

t' = (365 ¼ days)/0.436 = 837.7 days

^{½}t' = (365 ¼ days)/0.436 = 837.7 days

This is the time elapsed on the astronaut's clock when the Earth has made one revolution equal to 365 ¼ days. In other words, each of his days is roughly equal to 2.29 Earth days. Hence, his clock is obviously running

*slower*than the Earth clock.

A more intuitive way to look at the result would be in terms of the time transformation:

t
= t'/ (1 - v

^{2}/c^{2})^{½}
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