On an east-west shipping lane are ten ships sailing individually.  The first five from the west are sailing eastwards while the other five ships are sailing westwards.  They sail at the same constant speed at all times.  Whenever two ships meet, each turns around and sails in the opposite direction.  When all ships have returned to port, how many meetings of two ships have taken place?
1. **Understanding the problem**: We have ten ships on an east-west shipping lane. The first five ships from the west are sailing eastwards, and the other five ships from the east are sailing westwards. All ships sail at the same constant speed. When two ships meet, they turn around and sail in the opposite direction. We need to determine how many meetings of two ships take place before all ships return to port.

2. **Simplifying the problem**: Instead of considering the ships turning around, we can simplify the problem by imagining that they continue in their original directions. This simplification works because the relative motion and the number of meetings remain the same.

3. **Counting the meetings**: Each of the five ships sailing eastwards will meet each of the five ships sailing westwards. Therefore, we need to count the number of unique pairings between the eastward and westward ships.

4. **Calculating the number of meetings**: 
   - There are 5 ships sailing eastwards.
   - There are 5 ships sailing westwards.
   - Each eastward ship will meet each westward ship exactly once.

   Therefore, the total number of meetings is:
   \[
   5 \times 5 = 25
   \]

5. **Conclusion**: The total number of meetings of two ships is 25.

The final answer is \(\boxed{25}\).