This achievement is easy but can be tricky because the other planes will shoot down the UFOs if you're not careful.
The trick here is to pick a target and stay on the target until it's destroyed. The prologue isn't very long so be quick.
Some quick tips:
When you hit the brakes or use boost on your jet pack it doesn't seem to effect the your maneuverability at all. Which sucks because if you hit the brakes you should be able to turn faster which will give you a quicker bead on your target.
If you want this achievement early, then I'd suggest starting the game on Casual, getting this achievement and then starting over on a higher difficulty if you intend to play on Normal or Hardcore. It took me a couple of times on Hardcore to knock down 5 UFOs. The reason to start out on Casual is that new games overwrite previous saves. So start on Casual, get the achievement and then start a new game on a higher difficulty.