I bet he broke the sound barrier

(Spoilers for the Dark Knight Rises.)

At the end of the movie, Batman and his allies fail to stop the nuclear bomb, so he flies it out over the bay to prevent it from wiping out Gotham. People on the Internet correctly noted that this is definitely underestimating the range at which nuclear weapons can cause Bad Stuff to happen. But there definitely exists some distance such that the bomb going off at that range wouldn’t hurt anyone in Gotham to any instantly noticeable degree.

At the football scene, Dr. Scientist states that it’s a neutron bomb with a blast radius of six miles. Bane said it was a four-megaton bomb, and those numbers do not go well together. I’ll take the nuclear physicist’s word over Bane’s, and we can find out how far Batman must have gone.

I don’t fully understand why the equations I’m using are the right ones (I would have expected inverse squares instead of weird fractional exponents), but Wikipedia cited this as a source so it probably works. Anyway, for a neutron bomb (unlike most nuclear weapons) half the energy goes to radiation and does not affect the size of the blast radius. If the blast radius (in km) is (Yield/2.5kt)^.33, then a six-mile blast radius means 2410 kilotons. Double that, because half the energy is going into radiation instead of blast, and it’d take a 4.82 megaton bomb. So Bane wasn’t that far off after all.

We can determine how far Batman would have to fly it. It was obviously over six miles, or everyone watching it would have been blasted backward and very possibly killed just from the air. But it must have been even farther than that, because there were unprotected humans with a clear line of sight to the blast. When it went off, they cheered. They didn’t appear to have been covered in burns or struck blind or anything.

For a 4.8 Mt neutron bomb, the thermal radiation would hand out third degree burns within about a ten-mile radius. But the other effects of the thermal radiation reach go further than that.

The people on the bridge were all watching Batman fly the bomb away. And they were looking directly at it when it went off. (Note: Never do that.) Fortunately for them, it went off during the day instead of at night. Dilated pupils would be a bad thing. Unfortunately for them, it was a pretty clear day. A one-megaton nuke would temporarily blind people from 13 miles away. That’s for a regular nuclear weapon where 5% of the energy goes to radiation. For a neutron bomb, that number is 50%, leaving proportionately less for thermal radiation. (To make up for that, the neutron radiation is worse, but that has a smaller range anyway.)

Since a two-megaton neutron bomb would blind people from a bit over 13 miles, a 4.82-megaton one would blind people from 13√(4.82/2) miles away.  That’s a bit over 20 miles. Hopefully he flew it farther than that, but any closer and the kids in the school bus would definitely not have been cheering. So this gives us a good lower bound on his speed.

The timer on the bomb showed 1:57 when Batman attached it to his flying car. Then he kissed Catwoman and told Commissioner Gordon his not-remotely-secret-anymore identity and started the car, and by the time he took off it had been over 40 seconds. That leaves less than 77 seconds for him to fly it more than 20 miles. Apparently the Bat can fly at 935 miles an hour, which is well over the speed of sound. So I win my bet, muahaha.

Maybe there are other effects with wider reaches. Like, Gotham is probably going to have some severe fallout problems later. If there are effects that would show up on screen as soon as the bomb goes off and have a wider range than the flash blindness, Batman would have had to take the bomb even farther. But 20 miles in 77 seconds gives us a lower bound: He must have gone at least that fast.

(And it just occurred to me that I should be timing from when he passed the bridge instead of when he took off, since the bridge is where the people were. But the movie didn’t show the timer position for that, so I’ll just say it’s definitely significantly higher than 935 and use that number anyway.)

Incidentally, if he covered that much distance in that little time, then his average acceleration was 20/77*3600/77 miles per hour per second. Or 5.4 m/s^2, about half a gee. Which is much more survivable than I was expecting.
But the air resistance against the giant spherical bomb would be .5*1.225 kg/m^3*(418 m/s)^2*.47*π*.75^2 = 88.885 kiloNewtons. That’s about ten tons of force just from the air pushing back against the bomb. And the hovercraft can apparently fly at over 935 miles an hour while dragging that behind it. Too bad it got nuked, because that must be a seriously awesome machine.

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