A rocket is constructed on Earth with the following specifications: 96 m long, with a mass of 4.5 x 10^3 kg. a) On a trip to visit a nearby star...

Tuesday, February 17, 2009

A rocket is constructed on Earth with the following specifications: 96 m long, with a mass of 4.5 x 10^3 kg. a) On a trip to visit a nearby star...

Hello!

This question is about the special relativity theory. It states (and there are many experiments confirming it) that the speed of light in a vacuum is a constant, denoted $\displaystyle{c},$ regardless of the frame of reference. This gives some paradoxical results.

Denote the given speed of a rocket as $\displaystyle{v}.$ Then the Lorentz factor $\displaystyle\gamma=\sqrt{{{1}-\frac{{{{v}}^{{2}}}}{{{{c}}^{{2}}}}}}$ describes the changes in observed length, time and mass of a moving object. Namely, length is contracted, time is dilated and mass is increased with this factor. In numbers it is $\displaystyle\sqrt{{{1}-{{\left({0.68}\right)}}^{{2}}}}\approx{0.73}$ (dimensionless).

Therefore

a) the length will be $\displaystyle{96}\cdot\gamma\approx{70.4}{\left({m}\right)},$

b) the apparent mass will be $\displaystyle\frac{{{4.5}\cdot{{10}}^{{3}}}}{\gamma}\approx{6139}{\left({k}{g}\right)},$

c) the dilated time is given to be $\displaystyle{15}$ years, so the time for astronauts will be less by $\displaystyle\gamma,$ i.e. $\displaystyle{15}\cdot\gamma\approx{11}{\left({y}{e}{a}{r}{s}\right)}.$ Note that it is a simplification, because to return a rocket must change its velocity.

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Tuesday, February 17, 2009