Hitokage said:
"There isn't a way for them to change the direction of the object from heading away from us to heading towards us with the masses they are using."
"Considering that they are trying to discover the composition of the asteroid, and that the subsequent path of the asteroid could change radically after the impact, is this approach really the smartest idea?"
Yep, its the reading - she no works. The problem isn't with the change in direction, its the change in the surface. A comets course changes slightly overtime due to the emissions from its surface vents. If you don't know the internal composition of the object, you could fragment the inside of it creating fractures on the surface that would result in more and/or larger vents being opened on the surface.
So since you have a good understanding of physics, what happens to the core of a comet that is mostly a crystaline lattice as opposed to entirely solid. That's right - the inside can fracture. So go make a crystaline surface with some rock crystals and then wrap it with mud and dirt (one of the possible compositions of comets). Now hang it from the ceiling piniata style and throw the equivalent force at it. Now apply the same force (including heat) to that object as is applied by the sun and other celestial bodies on it.
Post pics of your results.
Oh, and don't forget to apply torsion to the object to account for that fractured inner surface - the source of the jets on the surface.