Two trains are moving with velocities v1 1 = 10 − ms and v2 1 = 20 − ms on the same track in opposite directions. After the application of brakes if their retarding rates are a1 2 = 2 − ms and a2 2 = 1 − ms respectively, then the minimum distance of separation between the trains to avoid collision is (a) 150 m (b) 225 m (c) 450 m (d) 300 m

To determine the minimum distance of separation between the two trains to avoid a collision, we need to analyze their motion before and after the brakes are applied. Let's break this down step by step.

Understanding the Motion of the Trains

We have two trains moving towards each other:

• Train 1 has an initial velocity of v1 = 10 m/s and a retarding rate of a1 = 2 m/s².
• Train 2 has an initial velocity of v2 = 20 m/s and a retarding rate of a2 = 1 m/s².

Calculating Stopping Distances

First, we need to calculate how far each train will travel before coming to a stop. The formula to find the stopping distance (d) when deceleration (a) is applied is:

d = v² / (2a)

Let's calculate the stopping distance for each train:

For Train 1:

Using the values:

• Initial velocity, v1 = 10 m/s
• Deceleration, a1 = 2 m/s²

Plugging into the formula:

d1 = (10 m/s)² / (2 * 2 m/s²) = 100 / 4 = 25 m

For Train 2:

Using the values:

• Initial velocity, v2 = 20 m/s
• Deceleration, a2 = 1 m/s²

Plugging into the formula:

d2 = (20 m/s)² / (2 * 1 m/s²) = 400 / 2 = 200 m

Finding the Minimum Distance of Separation

Now that we have the stopping distances for both trains, we can find the minimum distance of separation required to avoid a collision. Since the trains are moving towards each other, the total distance they need to cover before stopping is the sum of their stopping distances:

Minimum Distance = d1 + d2

Substituting the values we calculated:

Minimum Distance = 25 m + 200 m = 225 m

Conclusion

To ensure that the two trains do not collide, the minimum distance of separation required is 225 meters. Therefore, the correct answer is (b) 225 m.