A body lost in a jungle finds a note. In the note was written the following things.
Displacements
1. 300 m `53^(@)` South of East .
2. 400 m `37^(@)` North of East
3. 500 m North
4. `500sqrt(2) m ` North- West
5. 500 m South
He starts walking at constant speed 2m/s following these displacements in the given order.
How far and in which direction is he from the starting ponit after 5 min. and 50 s ?

Options

A. 500 m due East
B. 500 m due West
C. 700 m due South- West
D. 700 m due North-East

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Displacements
We have five displacements to consider:
1. **300 m South of East at 53°**
2. **400 m North of East at 37°**
3. **500 m North**
4. **500√2 m North-West**
5. **500 m South**

### Step 2: Break Down Each Displacement into Components
We will resolve each displacement into its x (East-West) and y (North-South) components.

1. **For 300 m South of East at 53°:**
- \( x_1 = 300 \cos(53°) \)
- \( y_1 = -300 \sin(53°) \)

2. **For 400 m North of East at 37°:**
- \( x_2 = 400 \cos(37°) \)
- \( y_2 = 400 \sin(37°) \)

3. **For 500 m North:**
- \( x_3 = 0 \)
- \( y_3 = 500 \)

4. **For 500√2 m North-West:**
- North-West means 45° from both North and West, so:
- \( x_4 = -500 \) (since it goes West)
- \( y_4 = 500 \) (since it goes North)

5. **For 500 m South:**
- \( x_5 = 0 \)
- \( y_5 = -500 \)

### Step 3: Calculate the Total Components
Now we sum up all the x and y components.

- Total x-component:
\[ X = x_1 + x_2 + x_3 + x_4 + x_5 \]

- Total y-component:
\[ Y = y_1 + y_2 + y_3 + y_4 + y_5 \]

### Step 4: Calculate the Individual Components
Using approximate values for trigonometric functions:
- \( \cos(53°) \approx 0.6 \), \( \sin(53°) \approx 0.8 \)
- \( \cos(37°) \approx 0.8 \), \( \sin(37°) \approx 0.6 \)

Calculating each component:
1. \( x_1 = 300 \times 0.6 = 180 \) m, \( y_1 = -300 \times 0.8 = -240 \) m
2. \( x_2 = 400 \times 0.8 = 320 \) m, \( y_2 = 400 \times 0.6 = 240 \) m
3. \( x_3 = 0 \), \( y_3 = 500 \)
4. \( x_4 = -500 \), \( y_4 = 500 \)
5. \( x_5 = 0 \), \( y_5 = -500 \)

### Step 5: Sum the Components
Total x-component:
\[ X = 180 + 320 + 0 - 500 + 0 = 0 \text{ m} \]

Total y-component:
\[ Y = -240 + 240 + 500 + 500 - 500 = 500 \text{ m} \]

### Step 6: Calculate the Resultant Displacement
The resultant displacement from the starting point is:
\[ R = \sqrt{X^2 + Y^2} = \sqrt{0^2 + 500^2} = 500 \text{ m} \]

### Step 7: Determine the Direction
Since the total x-component is 0 and the total y-component is positive, the direction is directly North.

### Final Answer
The body is **500 m North** from the starting point.

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