In the history of Earth, there have been five mass extinctions and arguably Humans themselves are the sixth mass extinction, causing species to die out at alarming rates and changing the climate so rapidly that many more are soon to follow. However, one of the most discussed extinction events is the asteroid impact 65 million years ago. The crater at Chicxulub, Yucatan is 180 kilometers across and 20 kilometers deep, taking the title of one of the largest impact events in the history of our planet with an asteroid of nearly 10 kilometers in diameter causing the most recent extinction event. Asteroid impacts are a very real threat and concern to humans. Objects the size of our moon float around in interplanetary space or the asteroid belt simply waiting for another stray rock to knock them out of their orbit and into ours. These rogue rocks and comets are known as Near Earth Objects, or NEOs and there are several organizations dedicated to finding, naming, tracking, and predicting these objects, and it's not a question of if we'll end up on one of these object's hit lists, but when.
There have been several proposed solutions to deflect or even destroy these objects before they can end up destroying our planet, including but not limited to; nuclear weapons, nuclear weapons, nuclear weapons, nuclear weapons, hitting it with something very big and fast to make it move slightly to the left, putting some sails on it and letting the sun push it away, attaching a probe to it and putting it into lunar orbit, and nuclear weapons. It seems to me that hitting it with a large object to change either it's speed or direction even the tiniest fraction to avoid a collision is the best course of action. One of my favorite science channels on YouTube, AsapSCIENCE, explains by way of guest host Bill Nye (the Science Guy) how changing the trajectory of an asteroid by altering it's speed or direction even the tiniest percent would work. Bill Nye (el Chico Ciencia) explains that an asteroid moving at 10 km/s toward Earth needs to be changed by 2 mm/s to avoid a collision. So a 10 kilometer asteroid is 10,000 meters, obviously, and one millimeter is one thousandth of a meter. So one thousand multiplied by ten thousand is ten million. So the object would need to be altered by one ten-millionth of it's original speed in order to avoid an impact, and of course, that's not very much. Yet, as Bill Nye (die Wissenschaft Kerl) points out; an asteroid with a mass of 100,000 tons is extremely difficult to move at all. It's going to take a lot of force to move something of that size even the tiniest fraction.
So as Bill Nye (le gars de la science) points out, the easiest and obviously the most scientific way of deflecting an asteroid is to take an enormous spacecraft and smash it into the side of the asteroid to change it's speed just a tiny bit. Now, the mass of a fully loaded Saturn V rocket is 2,970,000 kg, and during the third stage, the rocket is moving at 10.8 km/s. Assuming the asteroid with a mass of 100,000 tons (90909090.91 kg) is moving at 10 km/s, this means that the momentum of the asteroid is 909,090,909.1 kg km/s or 909,090,909,100 kg m/s and the momentum of the rocket is 32,076,000 kg km/s or 32,076,000,000 kg m/s, meaning the total momentum in the system is 941,166,909,100 kg m/s. If we divide the momentum of the rocket by the mass of the asteroid, we get a -.3528 m/s change in speed. This is far more than what is required to deflect the asteroid enough to avoid a world-ending collision.
Technically, this theory is feasible... If we could somehow get something with the mass of a fully loaded Saturn V rocket into space and on the correct course to slow the asteroid down .3528 m/s, though the size of a rocket needed to lift something the size of a Saturn V rocket into space would be unimaginably large for spacecraft standards, but still doable.
AsapSCIENCE video with Bill Nye (the Science Guy): https://www.youtube.com/watch?v=Agdvt9M3NJA