# Two moving cars (Kinematics problem)

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

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_{A}t – [itex]\frac{1}{2}[/itex]at^{2}

Then car B has the position v_{B}t + d

Now, to avoid collision, v_{B}t + d > v_{A}t – [itex]\frac{1}{2}[/itex]at^{2}

→ 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.

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