A man moves up on a 3 km long incline road with speed of 2m/s , then on a 3 km level road with speed of 3 m/s and moves down on a 3 km long incline road with 6 m/s. Find the average speed of man.

Answer

To find the average speed of the man moving along the different sections of the road, we can follow these steps:

### Step 1: Calculate Total Distance
The total distance covered by the man is the sum of the distances of all three segments of the journey:
- Distance up the incline: 3 km
- Distance on the level road: 3 km
- Distance down the incline: 3 km

Total distance \( S \) = \( 3 \, \text{km} + 3 \, \text{km} + 3 \, \text{km} = 9 \, \text{km} \)

### Step 2: Convert Total Distance to Meters
To work with consistent units, convert kilometers to meters:
\[ 9 \, \text{km} = 9 \times 1000 \, \text{m} = 9000 \, \text{m} \]

### Step 3: Calculate Time Taken for Each Segment
Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

1. **Time for the incline up (3 km at 2 m/s)**:
\[ t_1 = \frac{3000 \, \text{m}}{2 \, \text{m/s}} = 1500 \, \text{s} \]

2. **Time for the level road (3 km at 3 m/s)**:
\[ t_2 = \frac{3000 \, \text{m}}{3 \, \text{m/s}} = 1000 \, \text{s} \]

3. **Time for the incline down (3 km at 6 m/s)**:
\[ t_3 = \frac{3000 \, \text{m}}{6 \, \text{m/s}} = 500 \, \text{s} \]

### Step 4: Calculate Total Time
Now, sum up the times for all segments:
\[ \text{Total time} = t_1 + t_2 + t_3 = 1500 \, \text{s} + 1000 \, \text{s} + 500 \, \text{s} = 3000 \, \text{s} \]

### Step 5: Calculate Average Speed
The average speed \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values:
\[ V_{avg} = \frac{9000 \, \text{m}}{3000 \, \text{s}} = 3 \, \text{m/s} \]

### Final Answer
The average speed of the man is **3 m/s**.

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