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Supposing I had a planet the size of the earth, how much mass and how much velocity would an impacting body have to have to completely annihilate the planet? That it is to say that its reduced from a coherent planet to a shattered cloud of debris spread out over its former orbit, maybe a bit like an asteroid belt.
How would the impacting body compare to other objects in the solar system?
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The faster the body is moving, the smaller it can be to make this happen. on Physics.SE they had an answer for that.
the answer is 2.24 $\times$ 10$^{32}$J.
So you need to hit something with the total energy delivery of that amount.
An object about the size of the moon hit the Earth and the earth is still here. A large object such as Ceres traveling fast enough could annihilate the earth, and smaller object such as Palles or Vesta would also be to do the same job, if they are traveling a little faster than the previous one. But the faster it is traveling the smaller it needs to be. .9 light speed might only need something the size of a supertanker to do the job. My math isn't good enough to figure out the exact proportions for you though.
