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, we need to analyze the motion of each person and how they perceive the motion of others. We will break down the velocities step by step.

### Step 1: Determine the velocity of B with respect to A
- Person A observes B moving east with a speed of \(10 \, \text{m/s}\).
- Thus, the velocity of B with respect to A is:
\[ \vec{V}_{B/A} = 10 \hat{i} \]

### Step 2: Determine the velocity of C with respect to B
- Person B observes C moving south with a speed of \(20 \, \text{m/s}\).
- Therefore, the velocity of C with respect to B is:
\[ \vec{V}_{C/B} = -20 \hat{j} \]

### Step 3: Determine the velocity of D with respect to C
- Person C observes D moving west with a speed of \(30 \, \text{m/s}\).
- Hence, the velocity of D with respect to C is:
\[ \vec{V}_{D/C} = -30 \hat{i} \]

### Step 4: Determine the velocity of the tree with respect to D
- Person D observes a tree moving north with a speed of \(40 \, \text{m/s}\).
- Thus, the velocity of the tree (ground) with respect to D is:
\[ \vec{V}_{G/D} = 40 \hat{j} \]

### Step 5: Combine the velocities to find the velocity of A with respect to the ground
We will use the following relationship:
\[ \vec{V}_{A/G} = \vec{V}_{B/A} + \vec{V}_{C/B} + \vec{V}_{D/C} + \vec{V}_{G/D} \]

Substituting the values we found:
\[ \vec{V}_{A/G} = 10 \hat{i} + (-20 \hat{j}) + (-30 \hat{i}) + 40 \hat{j} \]
Now, combine the i and j components:
\[ \vec{V}_{A/G} = (10 - 30) \hat{i} + (-20 + 40) \hat{j} \]
\[ \vec{V}_{A/G} = -20 \hat{i} + 20 \hat{j} \]

### Step 6: Determine the actual direction of motion of A
The velocity vector \(\vec{V}_{A/G} = -20 \hat{i} + 20 \hat{j}\) indicates that A is moving in the direction of:
- West (negative x-direction) and North (positive y-direction).

To find the angle of this vector with respect to the positive x-axis (East direction), we can use the tangent function:
\[ \tan(\theta) = \frac{V_y}{V_x} = \frac{20}{-20} = -1 \]
This gives us:
\[ \theta = \tan^{-1}(-1) = -45^\circ \]
This angle indicates that A is moving in the direction of North-West.

### Final Answer
The actual direction of motion of person A is **North-West**.