Consider the solution of Example 2.2 from the standpoint of length contraction. Strategy The astronauts have only enough provisions for a trip lasting 16 years. Thus they expect to travel for 8 years each way. If the star system Alpha Centauri is 4.30 lightyears away, it may appear that they need

Question

Consider the solution of Example 2.2 from the standpoint of length contraction.

Strategy The astronauts have only enough provisions for a trip lasting 16 years. Thus they expect to travel for 8 years each way. If the star system Alpha Centauri is 4.30 lightyears away, it may appear that they need to travel at a velocity of 0.5c to make the trip. We want to consider this example as if the astronauts are at rest. Alpha Centauri will appear to be moving toward them, and the distance to the star system is length contracted. The distance measured by the astronauts will be less than 4.30 ly.

Accepted Answer

The contracted distance according to the astronauts in motion is [latex](4.30 ly ) \sqrt{1-v^{2} / c^{2}}[/latex]. The velocity they need to make this journey is the contracted distance divided by 8 years.

[latex]v=\frac{ ext { distance }}{ ext { time }}=\frac{(4.30 ly ) \sqrt{1-v^{2} / c^{2}}}{8 y }[/latex]

If we divide by c, we obtain

[latex]\beta=\frac{v}{c}=\frac{(4.30 ly ) \sqrt{1-v^{2} / c^{2}}}{c(8 y )}=\frac{(4.30 ly ) \sqrt{1-v^{2} / c^{2}}}{(8 ly )}[/latex]

[latex]8 \beta=4.30 \sqrt{1-\beta^{2}}[/latex]

which gives

[latex]\begin{aligned}&\beta=0.473 \&v=0.473 c\end{aligned}[/latex]

which is just what we found in the previous example. The effects of time dilation and length contraction give identical results.

Question 2.3: Consider the solution of Example 2.2 from the standpoint of ...

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