Two cars, A and B are traveling in same direction with velocities v_A and v_B. 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, v_A – v_B < [itex]\sqrt{2ad}[/itex]

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 v_At – [itex]\frac{1}{2}[/itex]at²
Then car B has the position v_Bt + d

Now, to avoid collision, v_Bt + d > v_At – [itex]\frac{1}{2}[/itex]at²
→ v_A – v_B < [itex]\frac{1}{2}[/itex]at + [itex]\frac{d}{t}[/itex]

Now I have to remove t from this equation.
Any help please?

Thanks in advance.