Two trains are moving with equal speed in opposite directions along two parallel railway tracks. If the wind is blowing with speed `u` along the track so that the relative velocities of the trains with respect to the wind are in the ratio `1:2,` then the speed of each train must be

Options

A. `3u`
B. `2u`
C. `5u`
D. `4u`

Explanation

To solve the problem, we need to analyze the situation involving two trains moving in opposite directions with respect to the wind. Let's break this down step by step.

### Step 1: Define Variables
Let the speed of each train be \( v \) and the speed of the wind be \( u \).

### Step 2: Determine Relative Velocities
- For Train 1 (moving to the right), the relative velocity with respect to the wind is:
\[ v_{1w} = v + u \]
- For Train 2 (moving to the left), the relative velocity with respect to the wind is:
\[ v_{2w} = v - u \]

### Step 3: Set Up the Ratio
According to the problem, the relative velocities of the trains with respect to the wind are in the ratio \( 1:2 \). Therefore, we can write:
\[ \frac{v_{2w}}{v_{1w}} = \frac{1}{2} \]
Substituting the expressions for \( v_{1w} \) and \( v_{2w} \):
\[ \frac{v - u}{v + u} = \frac{1}{2} \]

### Step 4: Cross-Multiply to Solve for \( v \)
Cross-multiplying gives us:
\[ 2(v - u) = 1(v + u) \]
Expanding both sides:
\[ 2v - 2u = v + u \]

### Step 5: Rearrange the Equation
Rearranging the equation to isolate \( v \):
\[ 2v - v = u + 2u \]
This simplifies to:
\[ v = 3u \]

### Step 6: Conclusion
Thus, the speed of each train is:
\[ v = 3u \]

### Final Answer
The speed of each train must be \( 3u \).

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