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The Earth would be broken into pieces if the total energy delivered by the impacts was comparable to the gravitational binding energy of the planet. Earth is a sphere, so its binding energy is $$U=...
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HDE 226868
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#1: Initial revision
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HDE 226868
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2020-07-06T21:20:37Z (over 4 years ago)
The Earth would be broken into pieces if the total energy delivered by the impacts was *comparable* to the [gravitational binding energy](https://en.wikipedia.org/wiki/Gravitational_binding_energy) of the planet. Earth is a sphere, so its binding energy is
$$U=\frac{3GM_{\oplus}^2}{5R_{\oplus}}=2.24\times10^{32}\text{ Joules}$$
[Randall says](https://what-if.xkcd.com/26/) that to properly slow down the planet's rotation, we'd need to bombard it with a dinosaur-killing-sized asteroid "every couple days" for ten years. [Schulte et al. 2013](https://ui.adsabs.harvard.edu/abs/2010Sci...327.1214S/abstract) place the energy of such an event at around $4\times10^{23}\text{ Joules}$; the asteroids would then deliver a total energy of
$$4\times10^{23}\text{ Joules}\times\frac{10\text{ years}}{2\text{ days}}=7.3\times10^{26}\text{ Joules}$$
which is about 300,000 times too low.
It's quite possible that the asteroids would ablate away much of the surface, and I imagine that the majority of species wouldn't make it through the first couple of months, but the planet itself would survive.