I. A cubic spacecraft of edge 1.0000 km (you are looking orthographically, from

Question

I. A cubic spacecraft of edge 1.0000 km (you are looking orthographically, from earth, at one of the faces) passes Earth moving eastward (in relation to you), at the speed of a pion whose kinetic energy is eight-and-one-eight times its rest energy. (a) Determine (to 4 significant figures) the speed of the spacecraft in units of c (the speed of light in vacuum) (b) According to you, what angle does the diagonal of the face you are looking at make with the easterly edge? (c) According to you, what is the mean lifetime of a tritium-3 nucleus on the captain's worktable, in the spacecraft, if a tritium-3 nucleus at rest has a mean lifetime of 17.79 years?

Explanation / Answer

a. As

gamma = 8 1/8 = 8.125

and

gamma = 1 / sqrt(1 - v^2/c^2) = 8.125

solving for v in terms of c,

v = 0.9924c [ANSWER]

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b.

The length along the east is contracted by a factor of 8.125. Thus, the easterly length is

x = 1.0000 km / 8.125 = 0.12307 km

Thus, the angle with respect to the east is

theta = Arctan (1.0000 km / 0.12307 km) = 82.983 degrees [ANSWER]

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C. The lifetime of the nucleus dilates by a factor of 8.125, thus,

Tapparent = 17.79 * 8.125 years = 144.54 years [ANSWER]

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