1. The problem statement, all variables and given/known data I have the following situation: car 1 is distance di ahead of car 2. Car 1 has either a constant speed v1 or accelerates with a1 (I eventually want to cover both cases). Car 2 accelerates faster than car 1. Car 2 is eventually going to catch up with car 1. I want to calculate the distance that car 2 can accelerate until it has to slow down to reach car 1's speed. The time/distance graph would look something like this: 2. Relevant equations d1 = v1 * t + 0.5 * a1 * t2 - di d2 = v2 * t + 0.5 * a2 * t2 3. The attempt at a solution I could calculate the time that it takes for car 2 to catch up with car 1 by combining the two equations above, because when car 2 has caught up to car 1, it will have travelled the same distance as car 1 (plus the initial difference in distance di). So: v1 * t + 0.5 * a1 * t2 - di = v2 * t + 0.5 * a2 * t2 When solving for t, I think I got the right time for the distances/speeds I put in my example calculation. However, now I need to determine at what distance car 2 would have to start slowing down in order to not crash into car 1, but line up behind it. I know at what rate car 2 will decelerate, but I don't know from what initial speed vi it will decelerate from (since car 2 is accelerating, this depends on when it starts decelerating), so I can't calculate how far the car will travel when decelerating. Is there a way to calculate this distance?