Be sure to show all calculations clearly and state your final answers in
complete sentences.
The Coral Model of Colonization. We can estimate the time it would take for a
civilization to colonize the galaxy. Imagine that a civilization sends
colonists to stars that are an average distance \(D\) away and sends them in
spacecraft that travel at speed \(v\). The time required for travel, \(t_{\text
{ravel }}\), is then \(t_{\text {travel }}=D / v\) Suppose that the colonists
build up their colony for a time \(t_{\mathrm{col}}\) at which point they send
out their own set of colonists to other star systems (with the same average
distance and same spacecraft speed). Then the speed at which the civilization
expands outward from the home star, \(v_{\mathrm{col}}\) (for the speed of
colonization), is \(v_{\mathrm{col}}=D /\left(t_{\text {travel
}}+t_{\mathrm{col}}\right) .\) However, this is true only if the colonization
is always directed straight outward from the home star. In reality, the
colonists will sometimes go to uncolonized star systems in other directions,
so we will introduce a constant \(k\) that accounts for this zigzag motion. Our
equation for the speed at which the civilization expands outward from the home
star is $$\begin{aligned}
v &=k \frac{D}{\left(t_{\text {travel }}+t_{\text {col }}\right)} \\
&=k \frac{D}{\left(\frac{D}{v}+t_{\text {col }}\right)}
\end{aligned}$$
For the purposes of this problem, assume that \(k=\frac{1}{2}\) and that the
average distance between star systems is \(D=5\) light-years.
a. How fast (as a fraction of the speed of light) does the civilization expand
if its spacecraft travel at \(0.1 c\) and each colony builds itself up for 150
years before sending out the next wave of colonists? How long would it take
the colonists to expand a distance of 100,000 light-years from their home star
at this rate?
b. Repeat part (a), but assume that the spacecraft travel at \(0.01 c\) and that
each colony builds itself up for 1000 years before sending out more colonists.
c. Repeat part (a), but assume that the spacecraft travel at \(0.25 c\) and that
each colony builds itself up for 50 years before sending out more colonists.