Time Pilot

As Vermin said, the bombs are tossed by the pilots. However, there is some more info that could be relevant. Firstly, the planes don't always drop bombs - for the most part they shoot small little shots since it's the first level and they want the game to be easy to start to get you adjusted. Therefore, you wont see any bombs being thrown really until you've been playing the level for about 5 minutes, or at least that's how it was in my experience. This also occurred if I happened to die (it would take 5 minutes from my new life for anyone to throw bombs).

When they toss them, they arc in a lob like fashion. Dive down so that you have slightly more time to shoot them, obviously making sure to avoid getting hit by said bombs. Remember, if you kill the boss (in the first level it's the zepplin), the level will end, so you'll want to avoid doing that until you've shot down all ten bombs. Don't try to combine this achievement with progressing through the game, since you're likely to accidently kill yourself as you prolong the level. The achievement will pop as you hit the 10th bomb I believe it was. Make sure to physically count how many you've shot down. Ones you run into counted for me if you're curious.