An express train is moving with a velocity `v_1`. Its driver finds another train is movig on the same track in the same direction with velocity `v_2`. To escape collision, driver applies a retardation a on the train. The minimum time of escaping collision be

Options

A. `t=(v_1-v_2)/(a)`
B. `t_1=(v_1^2-v_2^2)/(2)`
C. none
D. both

Explanation

To solve the problem of determining the minimum time required for the express train to escape a collision with another train moving in the same direction, we can follow these steps:

### Step 1: Identify the initial conditions
Let:
- \( v_1 \) = initial velocity of the express train
- \( v_2 \) = velocity of the other train
- \( a \) = retardation applied by the express train (note that retardation is a negative acceleration)

### Step 2: Write the equation for the final velocity
When the driver applies the brakes, the final velocity \( v_f \) of the express train after time \( t \) can be expressed using the equation of motion:
\[ v_f = v_1 - at \]
Here, \( at \) is the distance covered due to retardation.

### Step 3: Set the condition for avoiding collision
For the express train to avoid a collision with the other train, its final velocity \( v_f \) must be less than or equal to the velocity of the other train \( v_2 \):
\[ v_f \leq v_2 \]
Substituting the expression for \( v_f \):
\[ v_1 - at \leq v_2 \]

### Step 4: Rearrange the inequality
Rearranging the above inequality gives:
\[ v_1 - v_2 \leq at \]
This can be rewritten as:
\[ t \geq \frac{v_1 - v_2}{a} \]

### Step 5: Determine the minimum time
The minimum time \( t \) required to escape the collision is thus:
\[ t = \frac{v_1 - v_2}{a} \]
This equation gives us the minimum time needed for the express train to slow down sufficiently to avoid colliding with the other train.

### Final Answer
The minimum time to escape the collision is:
\[ t = \frac{v_1 - v_2}{a} \]

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