If we can do this, we can get a nuke on target. That doesn’t pulverize all the incoming mass, but it does redirect almost all of it (even if in chunks of various sizes) to missing the Earth if you hit it before it’s right on top of us. Only a slight change in the vector (now vectors) is needed.
Now, “right on top of us” does need definition. It depends on the size of the object, it’s speed, and to some degree the power of the nuke. For a fairly good size object, I’m guessing you need to hit it ~7 days out, to be on the safe side. So, you have to see it and determine it is a threat maybe 20 days out, assuming you have at least 4 nukes prepositioned in differing orbits (the most “optimum” one could be boosted to intercept course by essentially a large rocket 1st stage carrying it?)
Obviously, objects spotted further out can be mechanically splashed.
Of course, another threat is an object thought to only be coming close, that gets nudged by something too small to easily spot, on the way in.
> ... assuming you have at least 4 nukes prepositioned in differing orbits
I’d be willing to bet that the Chinese and Russians volunteer to launch their nukes up into a parking orbit.
A nuke would be like a firecracker in the vacuum of space. With no air to cause a shockwave, the only effects would be heat, radiation, and a small amount of shrapnel. No matter what the movies tried to portray, I don’t think nuclear warheads are the answer to an incoming asteroid. It will have to be a kinetic impact, which means a lot of time needed before impact with the Earth in which to nudge it off course.