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The plane's future position can be accurately estimated as soon as it spools-up on the runway. So the drones wouldn't need to catch up, just fly up to intercept. To ensure turbine destruction, the drones could have a payload of ceramic ball bearings, explosives, or both.

A passenger plane taking off is going to have a hard time seeing (let alone avoiding) such small, maneuverable craft.

Of course, this is a movie plot threat. Really, nobody is going to spend the time and resources to program drones like that.

People associate drones with multicopters. But really they're UAVs.. or.. you know, what you call drones, and can also be planes. small/cheap (800USD) electric powered planes can go about the same speed as an airliner at take off, that is, 400km/h. any faster and it needs to be turbine powered.

Also you'd go the opposite way (toward the plane) so the speed dont matter much...

Now, good luck to aim for the turbine - for all turbines, at once.

The drones you buy at RC shops, yes. Do a YouTube search for jet powered RC planes and you'll find several that exceed the takeoff speed of a commercial airliner.