Two ships A and B are `10 km` apart on a line running south to north. Ship A farther north is streaming west at `20 km//h` and ship B is streaming north at `20 km//h.` What is their distance of closest approach and how long do they take to reach it?

Explanation

To solve the problem of the two ships A and B, we will follow these steps:

### Step 1: Understand the positions and velocities of the ships
- Ship A is located farther north and is moving west at a speed of \(20 \, \text{km/h}\).
- Ship B is located south of ship A and is moving north at a speed of \(20 \, \text{km/h}\).
- The initial distance between the two ships is \(10 \, \text{km}\).

### Step 2: Set up the coordinate system
- Let’s place ship A at the coordinates (0, 10) (0 km east, 10 km north) and ship B at (0, 0) (0 km east, 0 km north).
- Ship A's velocity vector can be represented as \(\vec{V_A} = (-20, 0) \, \text{km/h}\) (moving west).
- Ship B's velocity vector can be represented as \(\vec{V_B} = (0, 20) \, \text{km/h}\) (moving north).

### Step 3: Calculate the relative velocity of ship B with respect to ship A
- The relative velocity \(\vec{V_{BA}}\) is given by:
\[ \vec{V_{BA}} = \vec{V_B} - \vec{V_A} = (0, 20) - (-20, 0) = (20, 20) \, \text{km/h} \]

### Step 4: Calculate the magnitude of the relative velocity
- The magnitude of the relative velocity is:
\[ |\vec{V_{BA}}| = \sqrt{(20)^2 + (20)^2} = \sqrt{400 + 400} = \sqrt{800} = 20\sqrt{2} \, \text{km/h} \]

### Step 5: Determine the angle of approach
- The angle \(\alpha\) between the relative velocity vector and the north direction can be calculated using:
\[ \tan(\alpha) = \frac{20}{20} = 1 \implies \alpha = 45^\circ \]

### Step 6: Find the closest distance of approach
- The closest distance of approach occurs when the line connecting the two ships is perpendicular to the direction of the relative velocity.
- This distance can be calculated using the sine of the angle:
\[ S_{\text{min}} = AB \cdot \sin(45^\circ) = 10 \cdot \frac{1}{\sqrt{2}} = 5\sqrt{2} \, \text{km} \]

### Step 7: Calculate the time taken to reach the closest approach
- The time \(t\) taken to reach the closest distance can be calculated using the formula:
\[ t = \frac{S_{\text{min}}}{|\vec{V_{BA}}|} = \frac{5\sqrt{2}}{20\sqrt{2}} = \frac{5}{20} = \frac{1}{4} \, \text{hours} = 15 \, \text{minutes} \]

### Final Answers:
- The closest distance of approach is \(5\sqrt{2} \, \text{km}\).
- The time taken to reach this distance is \(15 \, \text{minutes}\).

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