Be sure to show all calculations clearly and state your final answers in
complete sentences.
Long Trips at Constant Acceleration. Consider a spaceship on a long trip with
a constant acceleration of \(1 g\). Although the derivation is beyond the scope
of this book, it is possible to show that, as long as the ship is gone from
Earth for many years, the amount of time that passes on the spaceship during
the trip is approximately
\\[
t_{\text {ship }}=\frac{2 c}{g} \ln \left(\frac{g \times D}{c^{2}}\right)
\\]
where \(D\) is the distance to the destination and \(\ln\) is the natural
logarithm. If \(D\) is in meters, \(g=9.8 \mathrm{m} / \mathrm{s}^{2},\) and \(c=3
\times 10^{8} \mathrm{m} / \mathrm{s}\) the answer will be in units of seconds.
Use this formula as needed to answer the following questions. Be sure to
convert the distances from light-years to meters and final answers from
seconds to years; useful conversions: 1 light-year \(\approx 9.5 \times 10^{15}
\mathrm{m}\) \(1 \mathrm{yr} \approx 3.15 \times 10^{7} \mathrm{s}\).
a. Suppose the ship travels to a star that is 500 light-years away. How much
time will pass on the ship? Approximately how much time will pass on Earth?
Explain.
b. Suppose the ship travels to the center of the Milky Way Galaxy, about
28,000 light-years away. How much time will pass on the ship? Compare this to
the amount of time that passes on Earth.
c. The Andromeda Galaxy is about 2.2 million light-years away. Suppose you had
a spaceship that could constantly accelerate at \(1 g .\) Could you go to the
Andromeda Galaxy and back within your lifetime? Explain. If you could make the
journey, what would you find when you returned to Earth?