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-by-Step Solution:

1. **Identify the velocities of the trains:**
- Let the velocity of the express train (Train 1) be \( v_1 \).
- Let the velocity of the second train (Train 2) be \( v_2 \).

2. **Determine the relative velocity:**
- Since both trains are moving in the same direction, the relative velocity of Train 1 with respect to Train 2 is given by:
\[ v_{\text{relative}} = v_1 - v_2 \]

3. **Understand the need to avoid collision:**
- To avoid a collision, the relative velocity between the two trains must become zero. This means that Train 1 must decelerate to match the speed of Train 2.

4. **Apply the kinematic equation:**
- We can use the first equation of motion:
\[ v_f = v_i + at \]
- Here, \( v_f \) (final velocity) must be 0 (to avoid collision), \( v_i \) (initial relative velocity) is \( v_1 - v_2 \), and \( a \) is the retardation (which will be negative in this context).

5. **Substituting values:**
- Rearranging the equation gives:
\[ 0 = (v_1 - v_2) - at \]
- This can be rewritten as:
\[ at = v_1 - v_2 \]

6. **Solve for time \( t \):**
- Therefore, the time \( t \) required to avoid collision is:
\[ t = \frac{v_1 - v_2}{a} \]

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