It wouldn't be necessary to destroy the asteroid, only to deflect it. If that's done when it's far enough away, a well-aimed missile with no explosive would be enough.
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Nyx say: Blowing up an asteroid would be about the worse thing you could do. You are then turning a bullet into a bunch of shotgun pellets. The mass will still be the same, but now scattered over a wider area, potentially doing even more damage.
Besides, all the huge (as in ones that are over 100 meters in size) asteroids already have their orbits plotted. and none are any serious threat.
And if there was one that was really that big, it would be detected years before it would impact. So a possible method to deflect the asteroid would be to slow it down or speed it up by a few minutes, And the thing would then safely just drift on pass. You can do this by either placing a gravity tug in front of (or behind) the object, and slowly steer it away. Or, paint the object with a good coat of white paint, and changes it reflectivity. Then the solar photons would act as billions of little brakes, slowly the object down. And finally, by adding rockets to the object, stabilize its spin, then directing it away.
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