The driver of a train moving at a speed ` v_(1)` sights another train at a disane ` d`, ahead of him moving in the same direction with a slower speed ` v_(2)`. He applies the brakes and gives a constant teradation ` a` to his train. Show that here will be no collision if ` d gt (v_(1) -v_(2))^(2) //2 a`.

Options

A. `d gt ((upsilon_(1)-upsilon_(2))/(2alpha))`
B. `d lt ((upsilon_(1)-upsilon_(2))^(2))/(2alpha)`
C. `d gt ((upsilon_(1)-upsilon_(2))^(2))/(2alpha)`
D. None of these

Explanation

To solve the problem, we need to analyze the motion of the two trains and derive the condition under which there will be no collision.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- Let the speed of the first train (the one applying brakes) be \( v_1 \).
- Let the speed of the second train (the one ahead) be \( v_2 \) (where \( v_2 < v_1 \)).
- The distance between the two trains is \( d \).
- The first train applies brakes with a constant retardation \( a \).

2. **Relative Velocity**:
- The relative velocity of the first train with respect to the second train is given by:
\[ v_{\text{relative}} = v_1 - v_2 \]

3. **Using the Equation of Motion**:
- We can use the equation of motion which relates initial velocity, final velocity, acceleration, and distance:
\[ v^2 = u^2 + 2as \]
- Here, \( v \) will be the final velocity of the first train when it comes to rest, \( u \) is the initial relative velocity, \( a \) is the retardation (which will be negative), and \( s \) is the distance covered.

4. **Setting Up the Equation**:
- The first train will come to rest, so \( v = 0 \).
- The initial relative velocity \( u \) is \( v_1 - v_2 \).
- The retardation is \( -a \) (since it is deceleration).
- The distance \( s \) is \( d \).
- Substituting these into the equation gives:
\[ 0 = (v_1 - v_2)^2 - 2ad \]

5. **Rearranging the Equation**:
- Rearranging the above equation, we get:
\[ 2ad = (v_1 - v_2)^2 \]
- Dividing both sides by \( 2a \):
\[ d = \frac{(v_1 - v_2)^2}{2a} \]

6. **Condition for No Collision**:
- For there to be no collision, the distance \( d \) must be greater than the distance calculated above:
\[ d > \frac{(v_1 - v_2)^2}{2a} \]

### Final Result:
Thus, we have shown that there will be no collision if:
\[ d > \frac{(v_1 - v_2)^2}{2a} \]