A ship 'A' steams down to North at 16 km/h, and ship 'B' due west at 12 km/h. at a certain instant B is 10 km north east of A . find the magnitude of velocity of A relative to B?

Options

A. 10 kmph
B. 25 kmph
C. 6 kmph
D. 20 kmph

Answer

To find the magnitude of the velocity of ship A relative to ship B, we will follow these steps:

### Step 1: Define the velocities of the ships
- Ship A is moving north at a speed of 16 km/h. We can represent this as a vector:
\[ \vec{V_A} = 16 \hat{j} \text{ km/h} \]
- Ship B is moving west at a speed of 12 km/h. We can represent this as a vector:
\[ \vec{V_B} = -12 \hat{i} \text{ km/h} \]

### Step 2: Use the relative velocity formula
The relative velocity of ship A with respect to ship B is given by:
\[ \vec{V_{AB}} = \vec{V_A} - \vec{V_B} \]
Substituting the vectors we defined:
\[ \vec{V_{AB}} = 16 \hat{j} - (-12 \hat{i}) = 16 \hat{j} + 12 \hat{i} \]

### Step 3: Calculate the magnitude of the relative velocity vector
To find the magnitude of the vector \(\vec{V_{AB}}\), we use the Pythagorean theorem:
\[ |\vec{V_{AB}}| = \sqrt{(12)^2 + (16)^2} \]
Calculating the squares:
\[ |\vec{V_{AB}}| = \sqrt{144 + 256} = \sqrt{400} \]
Taking the square root:
\[ |\vec{V_{AB}}| = 20 \text{ km/h} \]

### Final Answer
The magnitude of the velocity of ship A relative to ship B is:
\[ \boxed{20 \text{ km/h}} \]