A man travels 200 km distance with the speed of 40 m/s and next 50 km with the speed of 10 m/s . Find the average speed of man.

Answer

To find the average speed of the man who travels different distances at different speeds, we can follow these steps:

### Step 1: Identify the distances and speeds
- The man travels:
- Distance \( S_1 = 200 \) km at speed \( V_1 = 40 \) m/s
- Distance \( S_2 = 50 \) km at speed \( V_2 = 10 \) m/s

### Step 2: Convert distances from kilometers to meters
- Since the speeds are given in meters per second, we need to convert the distances from kilometers to meters:
- \( S_1 = 200 \) km = \( 200 \times 1000 = 200000 \) m
- \( S_2 = 50 \) km = \( 50 \times 1000 = 50000 \) m

### Step 3: Calculate the total distance
- The total distance \( S \) traveled by the man is:
\[ S = S_1 + S_2 = 200000 \, \text{m} + 50000 \, \text{m} = 250000 \, \text{m} \]

### Step 4: Calculate the time taken for each segment of the journey
- The time taken for the first segment \( T_1 \) is calculated using the formula:
\[ T_1 = \frac{S_1}{V_1} = \frac{200000 \, \text{m}}{40 \, \text{m/s}} = 5000 \, \text{s} \]
- The time taken for the second segment \( T_2 \) is calculated similarly:
\[ T_2 = \frac{S_2}{V_2} = \frac{50000 \, \text{m}}{10 \, \text{m/s}} = 5000 \, \text{s} \]

### Step 5: Calculate the total time
- The total time \( T \) taken for the entire journey is:
\[ T = T_1 + T_2 = 5000 \, \text{s} + 5000 \, \text{s} = 10000 \, \text{s} \]

### Step 6: Calculate the average speed
- The average speed \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{S}{T} = \frac{250000 \, \text{m}}{10000 \, \text{s}} = 25 \, \text{m/s} \]

### Final Answer
The average speed of the man is \( 25 \, \text{m/s} \).

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