Two cars `1` and `2` move with velocities `v_(1)` and `v_(2)`, respectively, on a straight road in same direction When the cars are separated by a distance `d` the driver of car `1` applies brakes and the car moves with uniform retardation `a_(1)`, Simultaneously, car `2` starts accelerating with `a_(2)`, If `v_(1) lt v_(2)`, find the minimum initial separation between the cars to avoid collision between then.

Explanation

To solve the problem step by step, we will analyze the motion of both cars and derive the formula for the minimum initial separation \( d \) to avoid a collision.

### Step 1: Understand the motion of both cars
- Car 1 is moving with an initial velocity \( v_1 \) and applies brakes with a uniform retardation \( a_1 \).
- Car 2 is moving with an initial velocity \( v_2 \) and starts accelerating with an acceleration \( a_2 \).
- We know that \( v_1 < v_2 \).

### Step 2: Define the final velocities
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