Person A observes B moving in east direction with speed `10(m)/(s)`, B observes C moving in south direction with speed `20(m)/(s)`, C observes D moving in west direction with speed `30(m)/(s)` & D observes a tree moving with speed `40(m)/(s)` in north direction. Then the actual direction of motion of person `A` will be

Options

A. north-west
B. north-east
C. south-east
D. none of these

Explanation

To solve the problem step by step, we will analyze the motion of each person in relation to the others and determine the actual direction of motion of person A.

### Step 1: Define the directions and velocities
- Person A observes B moving east at a speed of 10 m/s.
- Person B observes C moving south at a speed of 20 m/s.
- Person C observes D moving west at a speed of 30 m/s.
- Person D observes a tree moving north at a speed of 40 m/s.

### Step 2: Set up the equations for relative velocities
1. **For A and B**:
\[ \vec{V}_{B} - \vec{V}_{A} = 10 \hat{i} \]
This implies:
\[ \vec{V}_{B} = \vec{V}_{A} + 10 \hat{i} \tag{1} \]

2. **For B and C**:
\[ \vec{V}_{C} - \vec{V}_{B} = -20 \hat{j} \]
This implies:
\[ \vec{V}_{C} = \vec{V}_{B} - 20 \hat{j} \tag{2} \]

3. **For C and D**:
\[ \vec{V}_{D} - \vec{V}_{C} = -30 \hat{i} \]
This implies:
\[ \vec{V}_{D} = \vec{V}_{C} - 30 \hat{i} \tag{3} \]

4. **For D and the tree**:
\[ \vec{V}_{\text{tree}} - \vec{V}_{D} = 40 \hat{j} \]
Since the tree is stationary, \(\vec{V}_{\text{tree}} = 0\):
\[ -\vec{V}_{D} = 40 \hat{j} \implies \vec{V}_{D} = -40 \hat{j} \tag{4} \]

### Step 3: Substitute and solve for velocities
- From equation (4), substitute \(\vec{V}_{D}\) into equation (3):
\[ -40 \hat{j} = \vec{V}_{C} - 30 \hat{i} \]
Rearranging gives:
\[ \vec{V}_{C} = -30 \hat{i} - 40 \hat{j} \tag{5} \]

- Now substitute \(\vec{V}_{C}\) into equation (2):
\[ -30 \hat{i} - 40 \hat{j} = \vec{V}_{B} - 20 \hat{j} \]
Rearranging gives:
\[ \vec{V}_{B} = -30 \hat{i} - 20 \hat{j} \tag{6} \]

- Finally, substitute \(\vec{V}_{B}\) into equation (1):
\[ -30 \hat{i} - 20 \hat{j} = \vec{V}_{A} + 10 \hat{i} \]
Rearranging gives:
\[ \vec{V}_{A} = -30 \hat{i} - 20 \hat{j} - 10 \hat{i} = -40 \hat{i} - 20 \hat{j} \]

### Step 4: Determine the direction of motion of A
The velocity vector of A is:
\[ \vec{V}_{A} = -40 \hat{i} - 20 \hat{j} \]
This means:
- The negative sign in front of \(\hat{i}\) indicates westward motion.
- The negative sign in front of \(\hat{j}\) indicates southward motion.

### Step 5: Find the resultant direction
To find the actual direction of motion, we can analyze the vector:
\[ \vec{V}_{A} = 40 \hat{i} + 20 \hat{j} \]
This vector points in the south-west direction.

### Conclusion
The actual direction of motion of person A is **South-East**.