This mission is promising for a scenario that is predictable and within space technology’s ability to affect a small asteroid. The problem is when a comet or other large body comes at us in an extremely eccentric orbit. Nothing can be done then. But we don’t have to worry, right? We’ve only got 10 years left cause of climate change.
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.
I am not sure why diverting a body on an eccentric orbit would be any more challenging than diverting one on a nearly circular orbit.
The change in position at nearest approach, to a first order, would be
Δx ~ Δv x T
where Δv is the vector change in velocity, T is time to go before nearest approach and vector Δx would be change in position at t = T.
A more interesting possibility would be the ability to direct an asteroid at the capitol of a military enemy, and hold them hostage to your good will and intentions. Imagine a weapon like that in the hands of Iran or Bill Gates.
It you can divert an asteroid to miss the earth, you can divert one to hit it.