It's a common belief that battleships moved sideways when they fire. I've talked with people who were aboard at the time who swear to it. But I'm an engineer, and thus believe the question can and should be answered by doing math.

Let's assume we're trying to get *Iowa* moving sideways as fast as possible by firing the guns. We'll take only a minimal load of fuel and ammo, and turn the ship into a hovercraft without adding any weight, so there's no water resistance to slow her down.^{1} Dulin & Garzke list *Iowa*'s 1943 light displacement as 44,579 metric tons.^{2} This is definitely low, but I'm willing to use it as an analytic assumption. Now, about the momentum of the broadside. Each AP shell weighs 2700 lb or 1224.7 kg. The muzzle velocity is 2500 ft/s or 762 m/s. A full 9-shell broadside thus has a total momentum of 8.4*10^6 kg*m/s, for a hovercraft side velocity of .1884 m/s, or .3662 kts. Or about 7.5 inches per second for those who don't instinctively think in knots.

But what about the propellant? There's 299.4 kg in each gun. How fast does it come out of the gun? Half of it comes out at about the projectile's velocity, while the other half comes out later and going somewhat faster, around 1,300 m/s. The total extra impulse comes out to 2.78*10^6 kg*m/s, which is enough to add .0623 m/s (.1211 kts/2.5 in/sec) to the ship's sideways speed.

Let's see if adding the 5" guns is enough to get our hovercraft battleship past the 1 ft/sec mark, or at least above half a knot. The heaviest 5" projectile was 24.7 kg, with 7.7 kg of propellant, with a muzzle velocity of 792.5 m/s. Firing all 10 5" guns on a broadside adds 2.76*10^5 kg*m/s. That comes to .0062 m/s, or .012 kts. Sadly, the .24 in/sec added by the 5" isn't enough to push us over the 1 ft/sec mark.

But what about roll, you ask. Is the ship in danger of capsizing when the guns are fired?

There are two ways to analyze this. First, let's assume that we treat *Iowa* as a rectangular block of uniform density in free space, with the dimensions of her hull.^{3} Assuming that the average line of effect of the guns is 62' above the keel, and that the ship is 53' high to the deck, for the light ship, the rotation rate after one salvo comes to 1.42°/sec.

This can be extended by treating the rotational energy of that roll into potential energy by integrating the righting moment, and assuming that it is conserved (which is a pretty bad assumption that will overestimate how much the ship rolls). This gives a maximum roll of 22.5°, which isn't quite enough to dip the disengaged railing below the waterline. The maximum dip is about 20', which is less than the freeboard amidships, but the results will be noticeable.

But what if the ship is heavier, in the condition we would actually expect the guns to be fired in? We'll use Dulin & Garzke's 1943 Optimum Battle displacement, of 56,314 metric tons, with a metacentric height of 7.68' (2.34 m). Using the same approximation given above, this produces a moment of inertia of 6.326*10^6 tons*m^{2}, and a roll rate of .823°/sec. The total rotational energy would then be 653.3 tons*m^2/s^2, and by integrating the righting force over the angles of roll, we get a maximum angle of heel of only 5.7° thanks to the greater GM and heavier ship. In practical terms, this would be noticeable, but not dramatically so.

The second approximation is even sillier. Assuming that we fire one broadside per second, and that the basic math behind metacentric height remains valid at large angles, what angle would the ship settle at? At light displacement, GM is 4.88', and the draft is 28' 8.25". Interestingly, in this condition, the ship actually does flip over. However, GM is very low in light condition because most of the removable weights, like fuel, are low down. So taking the next value given in Dulin & Garzke, we have a displacement of 56,314 metric tons, draft of 35', and GM of 7.68'. This gives an angle of heel of 45.7°. Obviously, the ship would not be stable at this angle, but that's not the biggest problem in this scenario. Firing the guns at 30 times their normal rate is.

From all of this, we can conclude that even a full broadside has little effect on the ship's motion. What looks in pictures like the ship moving sideways is in fact the effect of blast on the water and the ship's wake. Inside the ship, the flexing of non-armored bulkheads under blast pressure displaces air, creating the impression of movement. This is similar to how slamming a door can pop the cabinets open. There's probably slightly more effect in roll, but the ship would not be firing broadsides while light, and in practice firing-induced roll was often neglected in fire control, suggesting it wasn't that important.^{4}

1 This is not a serious proposal, and I would be most annoyed at anyone who actually did it. ⇑

2 Ships are traditionally measured in long tons of 2,240 lb. I use this everywhere on the blog, but metric tons are easier to do math on. ⇑

3 I'm well aware that this is a fairly terrible approximation, but it's easy to do. If you want a better one, be my guest. ⇑

4 A lot of the details in this post were cribbed from the canonical NavWeps writeup on the subject, although the rolling analysis is original to me, and I did my math at light displacement instead of standard. ⇑

## Comments

Does this have more to do with the concept being silly in the first place or due to the ship designers being good at their jobs?

I would say it's silly in the first place. First, battleships are frickin' huge compared to the projectiles they're throwing, and second, water resistance exists. A battleship's shape is not exactly ideal for moving sideways.

The analysis here doesn't rely on any special features in the ship's design, it's just comparing the momentum of the ship vs the momentum of the shot and saying "Yup, battleships are really big."

But a giant armored hovercraft would be so cool (eels optional)! Why do you have to be so dismissive of the best ideas?

@Gobbobobble

Beleester nailed it. It's all simple physics, nothing at all to do with the specifics of the design.

@Protagoras

It would be really cool if you did it to someone else's battleship. Mine is too pretty to ruin that way.

I'm pretty sure the winning move in this game is to put rotors on your aircraft carriers so they can fly about in the stratosphere. Mere hover-battleships are yesterday's weapon for yesterday's war.

An elegant weapon for a more civilised age.

I did more analysis on this, spurred on by recent discussion of this on SSC. I assumed that the rotational energy of the 1.42 deg/sec roll rate was entirely turned into potential energy produced when the ship rolled. I then worked out the potential energy form of the equation for righting moment from metacentric height by integrating. (It's fun to use calculus, so long as you aren't being graded.) Assuming that energy is conserved throughout (this is a really bad assumption) I got a maximum roll of 22.5 deg. This isn't quite enough to put the deck edge under (it's going to dip the unengaged side about 20' towards the sea, which is less the freeboard) but it will be noticeable.

However, it's worth pointing out that I did this with the lightest possible configuration for Iowa. More normal configurations are going to see a much larger metacentric height, which is going to cut how much the ship rolls a lot. More than that, a heavier ship is also going to see more righting arm, and lower rotational energy (because angular momentum isn't going to change, but a heavier ship means slower rotation, and rotational kinetic energy is dependent on the square of velocity). I doubt you'd get past 10 degrees in a practical configuration, although I'm not going to crunch the numbers tonight. And all of this assumes that no energy is lost to the ocean, which is the mother of all bad assumptions. But I can put a hard cap on how much it could be. It's certainly possible that crew high up in the ship could feel something, but it's definitely not the degree of effectiveness I've seen credited by a lot of former battleship crew.

@bean I would wager a lot of what crewmen are reporting is related to the jerk force. (The physics term, not a reference to any branch of the military that may or may not be present on the ship) Humans are particularly sensitive to jerk, and I would imagine it is a big factor in the feeling of the guns firing shipboard, and could be responsible for the overstatement of the other effects by the crew.

That's possible. It should be possible to get some idea of the acceleration on the ship from video of the guns firing. That doesn't necessarily tell us jerk, but at least gives some idea of what sort of numbers we're dealing with. (Also, do you have any quantifications of human sensitivity to acceleration/jerk? That might be interesting.)