The wording to this one is a little bit confusing. You need to make sure that not only you directly or indirectly destroy 3 cars in the span of 10 seconds, but that those 3 cars have never been hit by your car before that 10 second interval! Needless to say, that if you keep playing this mode normally, you drive as fast as you can, you will bump a lot, and since the first lap or two are easy to survive for human players and AI alike, you'd be hard pressed to do this naturally. Congrats if you did, but it requires a lot of luck - or, alternatively, careful planning. I identified two ideal solutions to pull it off, based on whether you have enough controllers or online buddies or playing solo.
1) Super-easy solution, requires 4 controllers or 3 buddies online:
Whether you're playing locally in split-screen with 4 players or by finding 3 other folks who want to get this achievement with you, the execution never changes. All human players stand still at the start, doing absolutely nothing. You can make sure to get closer to each other to make things easier, as long as you make sure there's no touching. As soon as the lead AI hits the first checkpoint, a 10 second countdown begins. When it hits 6 or so, make sure you touch all 3 cars slightly. All players will be destroyed in the span of 10 seconds by the timebomb. Achievement unlocked with little to no effort.
2) Medium difficulty solution for solo players:
Single Race, Jungle, Temple Ruins. Drive Or Explode, Expert, 7 laps. With this combination of difficulty and track, AI will be pushing and shoving each other a lot, causing many of them to go into red health as early as lap 1 or 2. Follow the pack from a safe distance, making sure you're always either last or that you have at least 3-4 cars ahead, and of course that you're never touching any other car. You will see drivers go low on health continously, but make sure you don't make a move if you're close to the finish line, which replenishes 35 health to all racers. When you see 3 cars that are on fire and with red health, hit them all in the span of 10 seconds, between the hits, the close walls, the speed loss and the other cars coming up there's good chances of them getting destroyed in a matter of seconds. It requires luck, because sometimes there's only 1 or 2 cars at a time go red on health. If you have even a single person helping you locally or online, you can make this one easier by having two players ram into merely 2 AIs, with then the second player going straight into a wall for an instadeath, after making sure you touched the car obviously. It still requires luck, but it just became notably easier. I'm reading people claim your own car counts towards the 3 counted for the achievement, but I don't believe this is the case, as I did not unlock the achievement destroying 2 cars and myself in 7-8 seconds.
A bit of a confusing achievement that requires you to play in an unnatural way, but it can be cheesed.