1. The problem statement, all variables and given/known data

Two cars, A and B are traveling in same direction with velocities vA and vB. When car A is distance d behind car B, the breaks on A are applied, causing a deceleration a. Show that to prevent a collision between A and B, it is necessary that, vA – vB < $\sqrt{2ad}$

2. Relevant equations

1D kinematics

3. The attempt at a solution

Suppose at the moment car A starts acceleration, it is at the origin, then car B is at x = d.
After time t, car A has position vAt – $\frac{1}{2}$at2
Then car B has the position vBt + d

Now, to avoid collision, vBt + d > vAt – $\frac{1}{2}$at2
→ vA – vB < $\frac{1}{2}$at + $\frac{d}{t}$

Now I have to remove t from this equation.