Three ships A, B and C are in motion. The motion of A as seen by B is with speed v towards north-east . The motion. Of B as seen by C is with speed v towards the north-west. Then as seen by A, C will be moving towards

Options

A. North
B. South
C. East
D. West

Answer

To solve the problem, we need to analyze the relative motions of the ships A, B, and C. We will break down the velocities and their directions step by step.

### Step 1: Understanding the Directions
- Ship A is moving towards B with a velocity \( v \) towards the North-East.
- Ship B is moving towards C with a velocity \( v \) towards the North-West.

### Step 2: Representing the Velocity of A with Respect to B
- The North-East direction can be represented as an angle of 45 degrees from both the North and East axes.
- Therefore, the velocity of A with respect to B can be expressed in vector form:
\[ \vec{v}_{AB} = v \cos(45^\circ) \hat{i} + v \sin(45^\circ) \hat{j} = \frac{v}{\sqrt{2}} \hat{i} + \frac{v}{\sqrt{2}} \hat{j} \]

### Step 3: Representing the Velocity of B with Respect to C
- The North-West direction can also be represented as an angle of 45 degrees, but this time it is directed towards the West and North.
- Therefore, the velocity of B with respect to C can be expressed as:
\[ \vec{v}_{BC} = -v \cos(45^\circ) \hat{i} + v \sin(45^\circ) \hat{j} = -\frac{v}{\sqrt{2}} \hat{i} + \frac{v}{\sqrt{2}} \hat{j} \]

### Step 4: Finding the Velocity of C with Respect to A
- To find the velocity of C with respect to A, we can use the formula:
\[ \vec{v}_{CA} = \vec{v}_{CB} + \vec{v}_{AB} \]
- Rearranging gives us:
\[ \vec{v}_{CA} = \vec{v}_{BC} + \vec{v}_{AB} \]

### Step 5: Adding the Vectors
- Now substituting the values:
\[ \vec{v}_{CA} = \left(-\frac{v}{\sqrt{2}} \hat{i} + \frac{v}{\sqrt{2}} \hat{j}\right) + \left(\frac{v}{\sqrt{2}} \hat{i} + \frac{v}{\sqrt{2}} \hat{j}\right) \]
- Simplifying this:
\[ \vec{v}_{CA} = \left(-\frac{v}{\sqrt{2}} + \frac{v}{\sqrt{2}}\right) \hat{i} + \left(\frac{v}{\sqrt{2}} + \frac{v}{\sqrt{2}}\right) \hat{j} \]
- The \( \hat{i} \) components cancel out:
\[ \vec{v}_{CA} = 0 \hat{i} + \left(2 \cdot \frac{v}{\sqrt{2}}\right) \hat{j} = \sqrt{2} v \hat{j} \]

### Step 6: Direction of C with Respect to A
- The positive \( \hat{j} \) direction indicates that C is moving towards the North as seen by A.
- However, since we need to find the direction of C as seen by A, we must consider the negative direction:
\[ \vec{v}_{CA} = -\sqrt{2} v \hat{j} \]
- This means C is moving towards the South as seen by A.

### Final Answer
As seen by A, C will be moving towards the South.