After reading, XKCD's what would happen if a baseball travelling at the speed of light (https://what-if.xkcd.com/1/) I'm curious as to how fast would a slug or baseball actually need to go at minimum to fuse air molecules.

He assumed that the baseball had a velocity of 0.99999 the speed of light. But what if it were 0.5 the speed of light, for example? Also, would there by any fusion if the "fast" baseball collided with a wall of heavier elements such as iron or cobalt instead of plain, old air?


1 Answer 1


In order to cause fusion the kinetic energy of a particle needs to overcome the Coulomb barrier $$V_c=\frac{e^2}{4\pi \epsilon_0}\frac{Z_1 z_2}{R_1+R_2}$$ where $Z$ is atomic charges, $R$ the nuclear radii, and $e$ is the electron charge. For hydrogen-hydrogen this is $0.96\times 10^{-13}$ J and for nitrogen-nitrogen $1.95\times 10^{-12}$ J.

A moving object presumably imparts its velocity $v$ on a boundary layer that then collides with air, so the relative velocity is $2v$ and the kinetic energy will be $(1/2)Av^2$ (the diatomic nature of the gas probably doesn't matter here, but maybe the kinetic energy is $Av^2$ if the other atom piles up behind the first one to really press it). So we get fusion when $(1/2)Av^2=V_c$ (with a negligible nudge downwards because of the Maxwell-Boltzmann velocity distribution of the gas that means some molecules will be faster and maybe a slightly bigger one for Maxwell-Jüttner distributed plasma around the projectile, but I will ignore these.)

So I get a "fusion speed" of $$v_{fusion}=\sqrt{2V_c/A}.$$

For hydrogen this is 3.57% c, for nitrogen 4.3% c.

This is (I think) not going to reach the Lawson criterion for fusion very well, so it is not going to be a major energy source for the ensuing mayhem.

  • $\begingroup$ Amazing answer, but I didn't understand why the relative velocity is 2v. $\endgroup$
    – Milind R
    Feb 4, 2022 at 19:33

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