Dude, doing the math yourself is NEVER wasted effort. Just make sure you have double checks in place.
Lots of things may be calculated. But reality is more complex than some easy equations. It would not be trivial to make contact at all. And if you shoot it, the effect may not be as predicted. One might not correctly know the shape or weight or faults or solidity of the asteroid. Small errors in estimation become large very easily.
No shame in getting any math question wrong. In fact, you usually learn more.
Increasing spin requires energy, where would that energy come from if the change of trajectory remained the same?
No it would change the trajectory less the a dead on shot. Since The impact would not act directly on the center of mass, but put both a force and torque on it.
No matter what, if you crash, most of your energy is going to go into heat. If some energy goes into spin, then that just means a little less goes into heat. Momentum is conserved regardless.
If you look at it as a idealized case (no heat) you can see that this angular momentum has to come from somewhere, it has to be conserved or we break physics. This is because the projectile, upon impact, even if not rotating, is rotating in regards to the axis around the center of mass of the astroid. So again momentum is conserved AND angular momentum is conserved and energy is conserved, a off centered shot will impart less delta V then a dead on shot because of this.
There is no “idealized case (no heat)”, because all of that energy has to go somewhere, and conservation of momentum means that most of it can’t go into changing the trajectory. Heat, and maybe spin, are the only places the energy can go.
So, if we have rocket with linear momentum p and asteroid with linear momentum P, these two objects will, collectively, have momentum (p+P) This value will not change, regardless of how the two objects interact with each other, whether it’s a perfect hit or a perfect miss, or something in between. This collective momentum can only change if some outside object interacts with one or both of the objects.
Thus, if the rocket is… fully absorbed… by the asteroid, whether by a direct hit or an off center blow, the asteroid (now asteroid+rocket) will have momentum (p+P). This will change if part of the rocket, or asteroid, shears off and flies away.
For my money, I’d go for a head on impact. Hitting a high speed object from the side is hard in the best of circumstances
Are you sure about that?
But you want to hit it from the side. The whole idea is to deflect the course of the asteroid. The sooner you hit it, the further from earth) the more the trajectory with change. You are deflecting the path a certain amount based on the sideways momentum imparted to the asteroid. That angle causes it to diverge further from original course each minute. The question is how much you divert the path - enough to miss? If you hit it around Mar’s orbit, probably will have good results. Wait until it’s inside the moon’s orbit and you better use every nuclear warhead on earth.
Another factor is - where is the rock projected to hit? Unless it’s dead center, you have a head start- the rock is already close to missing. If it’s hitting the leading edge of earth deflect more forward - a bit further forward may cause it to miss. If it’s aiming for Greenland, deflect the trajectory upward toward the north pole. Yes, hitting it head-on to slow it may also have the desired effect. It’s all about the physics. Slowing a meteor aimed at the leading edge of earth simply makes it more likely to hit dead center unless you slow it enough; if it’s going to hit the trailing edge of earth, slowing down is an OK option.
Also note - the meteor that hit northern Russia, caught on multiple security and dash cams, was supposedly the size of a 5-story building. Based on the lack of impact crater it’s believed it was made of ice, IIRC. However, it apparently shattered windows for hundreds of miles around. So the last thing you want is unpredictable numbers of fragments the size of multi-story buildings. heading to who knows what part of earth. Fragging is a bad idea… and you have no good idea what any impact from any direction will do to the asteroid’s structural integrity.
I would imagine that modern targeting software can properly direct a side impact on a moving target accurately.
Of course, no one wants any problems but mass extinctions or years of crop failure are worse than replacing a few square miles of structure.
Presumably, if you could be accurate aiming at something at Mars, which I doubt, you could get it several times. Don’t see why the moon is too late.
I wonder what is the smallest asteroid they could even identify. And how far out. And the chances of making the identification at all for a given size. How does this work?
Because the asteroid is going pretty damn fast in relation to Terran orbit.
Maybe. But presumably if you have the accuracy to get it at Mars, it would be easier still if it was much closer regardless of its speed. It’s probably a bad comparison, but it still takes a rocket 8.5h to fly to the moon.
It took the Apollo program some three days to get to lunar orbit.
The further away an asteroid is, the less amount of energy is required to change it’s path.
I hadn’t thought of that.
Check out in and around 1:08 on the video, no they don’t go to the “exact same height” as claimed, the spinning one is clearly, but marginally lower. Now that could be in the margin of error for that experiment, but it does clearly show that the off centered shot is lower then the dead on shot.
Are you sure about that?
It’s not like firing a rifle at a target. Your projectile will have sensors and rockets on it, so it can correct its course en route. Hitting the target is about equally easy no matter how far away it is. Likewise, a side-on hit is about as easy as a head-on.
It’s like, when an Australian businessman flies to the US for a business meeting, and shakes the American businessman’s hand, he doesn’t miss the handshake just because he traveled tens of thousands of miles to do it.
Any idea the furthest away you could target? The smallest size you could identify? If something is large enough to take out Florida, and you can’t see it until too late, the sensors don’t matter.
And while it may take less energy to change a path further out, I’m not convinced this matters much.
If the Australian Prime Minister flies to America to shake Trump’s hand, it may not happen despite the fact he has travelled thousands of miles. Or it may be one of those awkward dominance three minute deals. 
We could detect such a rock at the distance of Pluto, but that’s not really the right question, because we probably wouldn’t. To do that, you have to point your telescope in the right direction. And since you don’t know in advance what the right direction is, practically speaking, that means you need to just sweep out the whole sky. And that’s not something we’re doing right now, which has led to some uncomfortably close calls (like, rocks big enough to cause mass extinctions, that got closer than the Moon, that we didn’t even notice until afterwards).
As for distance at which we can hit a rock, there’s no meaningful limit. If aliens told us about a rock orbiting a star in a completely different arm of the Galaxy, and we wanted to hit it, the only limit would be that our spacecraft wouldn’t survive long enough to make the trip. But the navigation wouldn’t be a problem.