A train goes from P to Q with an average speed of u km/h and returns from Q to P with an average speed of v km/h. What is its average speed for the entire journey in km/h?

Options

A. `(2uv)/(u+v)`
B. `sqrt((u^(2)+v^(2))/(2))`
C. `(u+v)/(2)`
D. `sqrt(uv)`

Answer

To find the average speed for the entire journey of the train traveling from P to Q and back from Q to P, we can follow these steps:

### Step 1: Define the distances and speeds
Let the distance from P to Q be \( x \) km. The average speed from P to Q is \( u \) km/h, and the average speed from Q to P is \( v \) km/h.

### Step 2: Calculate the total distance
The total distance for the entire journey (from P to Q and back to P) is:
\[ \text{Total distance} = x + x = 2x \text{ km} \]

### Step 3: Calculate the time taken for each leg of the journey
- Time taken to travel from P to Q:
\[ \text{Time}_{PQ} = \frac{x}{u} \text{ hours} \]
- Time taken to travel from Q to P:
\[ \text{Time}_{QP} = \frac{x}{v} \text{ hours} \]

### Step 4: Calculate the total time for the entire journey
The total time for the journey is the sum of the time taken for each leg:
\[ \text{Total time} = \text{Time}_{PQ} + \text{Time}_{QP} = \frac{x}{u} + \frac{x}{v} \]

### Step 5: Simplify the total time
To combine the times, we find a common denominator:
\[ \text{Total time} = \frac{xv + xu}{uv} = \frac{x(u + v)}{uv} \text{ hours} \]

### Step 6: Calculate the average speed
The average speed for the entire journey is given by the formula:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} \]
Substituting the values we calculated:
\[ \text{Average speed} = \frac{2x}{\frac{x(u + v)}{uv}} = \frac{2x \cdot uv}{x(u + v)} \]
The \( x \) cancels out:
\[ \text{Average speed} = \frac{2uv}{u + v} \text{ km/h} \]

### Final Answer
The average speed for the entire journey is:
\[ \frac{2uv}{u + v} \text{ km/h} \]

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