For example, ship $B$ is located $45^{\circ}$ north of ship $A$, and the distance between the two ships is $10 \sqrt{2} \mathrm{~km}$. If ship $A$ sails west and ship $B$ sails south at the same time, with ship $B$'s speed being twice that of ship $A$, what is the closest distance between ships $A$ and $B$ in kilometers?
Solution: As shown in Figure 3, let after $t$ hours, ships $A$ and $B$ have sailed to positions $A_{1}$ and $B_{1}$, respectively, and let $A A_{1}=x$. Thus, $B B_{1}=2 x$.

Given $A B=10 \sqrt{2}$, we have $A C=B C=10$. $\therefore A_{1} C=|10-x|, B_{1} C=|10-2 x|$.
Therefore, $A_{1} B_{1}=\sqrt{|10-x|^{2}+|10-2 x|^{2}}$ $=\sqrt{5(x-6)^{2}+20}$.

Hence, when $x=6$, $A_{1} B_{1}=2 \sqrt{5}$ is the minimum.