# What's the largest mushroom cloud possible from a coffee cup/grenade sized nuclear bomb?

Assuming the coffee cup is $16$oz = $1$lb = $0.4536$kg or $\sim 450$mL

I did a quick comparison to the W54 assuming there was a linear ratio (wishful thinking perhaps), and got it to be around 8m high.

Height of mushroom cloud: $1450$ ft

Weight of bomb: $58.1$ lb

Volume of bomb: $5.85$ inches in diameter, $15$ inches long

Source, see Socorro test

Assuming comparable height-to-weight ratio:

Mushroom cloud height of $16$ oz ($1$ lb) bomb = $1450$ft / $58.1$ lb $\times$ $1$lb $\sim 7.6$m

Assuming comparable height-to-volume ratio:

Cloud height of $16$ oz ($454$ mL) bomb = $1450$ft / $(\pi 5.85”^2 \times 15”) \times 454$ mL $\sim 8.3$m

But I'm curious what a more realistic value might be, assuming it was designed for that size and assuming maximum efficiency. Thanks in advance.

• ....Are you making a suitcase bomb? Nov 11 '14 at 21:20
• Haha, much more trivial. Proposing an awareness campaign for the Comprehensive Test Ban Treaty by putting trivia on coffee sleeves and a mushroom cloud model. Nov 11 '14 at 21:23
• Are we allowed to expend arbitrary amounts of energy to briefly compress a mass to cup size? Nov 11 '14 at 23:05
• @crclayton I'm almost certain that $1450 ft$ is just the height of the bomb when it went off!
– user12029
Nov 12 '14 at 0:05
• @NeuroFuzzy Oh, damn, maybe you're right. Nov 12 '14 at 3:26

Of all the common nuclear fuels, Pu-239 has the smallest critical mass. A spherical untampered critical mass is about 11 kg (24.2 lbs),1 10.2 cm (4") in diameter. Using appropriate triggers, neutron reflectors, implosion geometry and tampers, this critical mass can be reduced by more than twofold. This optimization usually requires a large nuclear development organization supported by a sovereign nation.

I measured my coffee cup, and it is clearly smaller in volume than the volume of a spherical critical mass of plutonium. Using all the other fancy hardware described in the second sentence of the quote would reduce the amount of plutonium needed, but would presumably increase the total size of the bomb.

Therefore I think the answer is that the largest possible explosion from a holy nuclear hand grenade is zero, for any common nuclear fuel.

I believe I heard somewhere that according to rough estimates, substances from the nuclear island of stability might have critical masses as small as a pencil eraser. Luckily for our civilization, there is probably no practical way to make bulk quantities of these atoms.

Hirosima was 15kt and had a volume of $1.5m^3$, it produced a mushroom cloud of about $17000m$

if things scaled linearly the coffee mug would produce a cloud of $0.0005m$ in height

However I really doubt it would be linear, even on the site that you linked there doesn't seem to be much of a correlation between kt and mushroom cloud height, not to mention there are multiple types of clouds each of which has varying heights.

http://nuclearsecrecy.com/nukemap/

• Yeah, I did similar comparisons to a bunch of bombs like Tsar Bomba and Ivy Mike, all with underwhelming results. Note that Hiroshima wasn't 15kT though, it had a 15kT TNT equivalent. Now I'm more curious about the largest theoretical yield assuming 100% efficiency of the fissile material and such, which is beyond my physics level. Nov 11 '14 at 21:54
• @crclayton Peak observed efficiency for fission of fissile atoms in nuclear weapons is worse than typical thermodynamic efficiencies for heat engines. 100% fission cannot be reached before the assembly becomes subcritical. Nov 13 '14 at 0:12
• I doubt that the Hirosima bomb contained 1.5 cubic meters of fissile material. Jun 26 '20 at 14:55