Two car `A` and `B` travelling in the same direction with velocities `v_1` and `v_2(v_1 gt v_2)`. When the car `A` is at a distance `d` ahead of the car `B`, the driver of the car `A` applied the brake producing a uniform retardation `a`. There wil be no collision when.

Options

A. `d lt ((v_1 - v_2)^2)/(2 a)`
B. `d lt (v_1^2 - v_2^2)/(2a)`
C. `d gt ((v_1 - v_2)^2)/(2 a)`
D. `d gt (v_1^2 - v_2^2)/(2 a)`

Explanation

To solve the problem of determining when there will be no collision between cars A and B, we can follow these steps:

### Step 1: Understand the Initial Conditions
Car A is ahead of car B by a distance \( d \). The velocities of the cars are \( v_1 \) for car A and \( v_2 \) for car B, with the condition that \( v_1 > v_2 \). When car A applies the brakes, it experiences a uniform retardation \( a \).

### Step 2: Determine the Relative Velocity
The initial relative velocity between car A and car B is given by:
\[ v_{\text{relative}} = v_1 - v_2 \]
This represents how fast car A is moving away from car B initially.

### Step 3: Analyze the Motion of Car A
When car A applies the brakes, it will decelerate. The final velocity of car A when it comes to a stop can be calculated using the equation of motion:
\[ v^2 = u^2 + 2as \]
Here, \( v = 0 \) (final velocity), \( u = v_1 \) (initial velocity), \( a = -a \) (retardation), and \( s \) is the distance traveled by car A before it stops.

Rearranging gives:
\[ 0 = v_1^2 - 2as \implies s = \frac{v_1^2}{2a} \]
This distance \( s \) is how far car A travels before it stops.

### Step 4: Determine the Distance Covered by Car B
During the time car A is stopping, car B continues to move. The distance covered by car B while car A is stopping is given by:
\[ \text{Distance covered by B} = v_2 \cdot t \]
where \( t \) is the time taken for car A to stop.

### Step 5: Calculate the Time Taken for Car A to Stop
The time \( t \) taken for car A to stop can be calculated using:
\[ v = u + at \implies 0 = v_1 - at \implies t = \frac{v_1}{a} \]

### Step 6: Substitute Time into Distance Covered by Car B
Substituting \( t \) into the distance covered by car B gives:
\[ \text{Distance covered by B} = v_2 \cdot \frac{v_1}{a} \]

### Step 7: Set Up the Condition for No Collision
For there to be no collision, the distance between the two cars after car A stops must be greater than the initial distance \( d \):
\[ d + \text{Distance covered by B} > s \]
Substituting the expressions we derived:
\[ d + v_2 \cdot \frac{v_1}{a} > \frac{v_1^2}{2a} \]

### Step 8: Rearranging the Inequality
Rearranging gives:
\[ d > \frac{v_1^2}{2a} - v_2 \cdot \frac{v_1}{a} \]
This simplifies to:
\[ d > \frac{v_1^2 - 2v_1v_2}{2a} \]

### Step 9: Final Condition for No Collision
Thus, the condition for no collision can be expressed as:
\[ d > \frac{(v_1 - v_2)^2}{2a} \]

### Conclusion
The final condition for no collision is:
\[ d > \frac{(v_1 - v_2)^2}{2a} \]