(About) 67 million years ago, space thought to itself "Y'know? That little life-sustaining planet 'Earth' is just doing too well for itself. I'm gonna go shake things up a little bit!" and hurled a 10km wide asteroid at our poor little home. All storytelling aspects aside, I was thinking, "If we generate a relatively tiny little bit of momentum that ever-so-slightly pushes the Earth when we jump up and land, what type of momentum would a massive collision do? When I first embarked on this endeavor, I decided that I could work to some answers using a variation of the conservation of momentum equation. Here are the values I used in my in calculations. I am treating Earth as a stationary reference frame, and the collision as perfectly inelastic:
mEarth = 5.972 x 1024 kg
mAsteroid = 6 x 1014 kg[i]
vAsteroid = 32,000 m/s[ii]
mEarth.2 = 5.972 x 1024 kg – 7 x 1010 kg[iii]
One thing I originally didn’t take into consideration was the value I have listed as “mEarth.2” which is the mass of Earth after losing approximately 70 billion tons of rock and dust as debris from the collision. Will it make that much of a difference in our calculations? No, as even that massive amount is tiny relative to the whole Earth, but I still thought it interesting to include. However, the article that I obtained that value from discussed where that rock went and offered some fascinating suggestions that these rocks may have carried life to Mars and even Jupiter’s moon Europa. Feel free to check it out!
So anyways, using (mAsteroid x vAsteroid) = (mAsteroid + mEarth.2)vf , you could find a final velocity of the affected Earth to be 3 x 10-6 m/s. So even for such a catastrophic collision, the mass of the earth keeps it from being fully effected.
To acknowledge some shortfalls in my calculation, I know that the Earth is also constantly moving in both its axial rotation and orbit, so this would affect the momentum calculation. Moreover, the rough approximation for the masses also makes the calculation inaccurate, as a lot of the rock would have been vaporized upon impact, and the larger chunks that did travel away from the Earth would have their own momentum as well. Overall however, I had a lot of fun with this calculation and it still blows my mind how hard of a hit the Earth can take!