Two trains are moving with velocities `v_1 = 10 ms^-1 and v_2 = 20 ms^-1` on the same track in opposite directions. After the application of brakes if their retarding rates are `a_1 = 2 ms^-2 and a_2 = 1 ms^-2` respectively, then the minimum distance of separation between the trains to avoid collision is

Options

A. 150 m
B. 225 m
C. 450 m
D. 300 m

Explanation

To find the minimum distance of separation between two trains moving towards each other to avoid a collision, we can follow these steps:

### Step 1: Identify the Given Values
- Velocity of Train 1, \( v_1 = 10 \, \text{m/s} \)
- Velocity of Train 2, \( v_2 = 20 \, \text{m/s} \)
- Retardation of Train 1, \( a_1 = 2 \, \text{m/s}^2 \)
- Retardation of Train 2, \( a_2 = 1 \, \text{m/s}^2 \)

### Step 2: Use the Kinematic Equation
We need to find the distance each train travels before coming to a stop. We can use the kinematic equation:
\[ v^2 = u^2 + 2as \]
where:
- \( v \) is the final velocity (0 m/s, since the trains come to a stop),
- \( u \) is the initial velocity,
- \( a \) is the acceleration (negative for retardation),
- \( s \) is the distance traveled.

### Step 3: Calculate the Stopping Distance for Train 1
For Train 1:
- Initial velocity, \( u_1 = 10 \, \text{m/s} \)
- Final velocity, \( v_1 = 0 \, \text{m/s} \)
- Retardation, \( a_1 = -2 \, \text{m/s}^2 \)

Using the equation:
\[ 0 = (10)^2 + 2(-2)s_1 \]
Rearranging gives:
\[ s_1 = \frac{10^2}{2 \times 2} = \frac{100}{4} = 25 \, \text{m} \]

### Step 4: Calculate the Stopping Distance for Train 2
For Train 2:
- Initial velocity, \( u_2 = 20 \, \text{m/s} \)
- Final velocity, \( v_2 = 0 \, \text{m/s} \)
- Retardation, \( a_2 = -1 \, \text{m/s}^2 \)

Using the equation:
\[ 0 = (20)^2 + 2(-1)s_2 \]
Rearranging gives:
\[ s_2 = \frac{20^2}{2 \times 1} = \frac{400}{2} = 200 \, \text{m} \]

### Step 5: Calculate the Minimum Distance of Separation
The minimum distance of separation \( d \) to avoid collision is the sum of the distances traveled by both trains:
\[ d = s_1 + s_2 = 25 \, \text{m} + 200 \, \text{m} = 225 \, \text{m} \]

### Final Answer
The minimum distance of separation between the trains to avoid collision is \( 225 \, \text{m} \).

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