I was equal parts grossed out and astonished (ok, maybe a little more grossed out than astonished) when I watched this video of a sperm whale exploding. Warning: this is a video of a sperm whale exploding. Obviously, it's not going to be pretty.

And being a physics geek, the first question that popped into my head was, "I wonder how much pressure built up inside that whale for it to explode like that?"

First, what's up with these exploding whales anyway? In an interview with Fusion, here's how deep sea ecologist Andrew David Thaler explains this amazing and unsightly phenomenon.

A whale is a really nice, contained package with a big, big layer of blubber around it that’s designed to keep everything in and keep water out while it’s diving. So they actually make fairly good balloons [..] And, like most mammals, when they die and they aren’t scavenged—-or they’re too big to be effectively scavenged—-their viscera begins to decompose, whatever contents were in their stomach. That produces methane and hydrogen sulphide and a couple other gases, which are going to begin expanding, especially if it’s sitting in the sun for a couple of weeks. Eventually, they can explode.

So let's make a rough, back-of-the-envelope style of calculation, to estimate the pressure inside a bloated beached whale that's about to explode. The idea is to get a ballpark estimate, that's within an order of magnitude of the actual result.

Why, you may ask? Because SCIENCE. That's why.

Here's the game plan. I'm going to open up the above video in the handy physics video analysis software Tracker.

Let's guess that the person in the video is about 1.8 meters tall (~ 6 feet), which is the average height for a Dutch Danish man. (The brave soul in the video is marine biologist Bjarni Mikkelsen, and the whale was beached on the Faroe islands.) This sets a scale for the other distances in the video.

Now, I'm going to track the speed at which the blood and gas mixture shoots out of the whale (the liquid shoots out first, and the guts follow after).

I tracked this explosion along 4 different paths. Averaging these four values gives me an average speed of 17.7 meters/second (with a standard deviation of 3.4 meters/second).

Boom! The blood shoots outs of the whale at a whopping 17.7 meters/second (or about 40 mph)!

Just for fun, here's a random question. If this fountain of blood was aimed straight up, how high would it reach? To work this out, we need to use the law of conservation of energy. When launched out of the whale, a drop of blood has a bunch of kinetic energy, and at the very top of its trajectory, all this kinetic energy is converted into its gravitational potential energy. So we can set these two quantities equal to each other, and solve for the maximum height that the blood can reach.

In this equation, v is speed, g is the gravitational acceleration of 9.8 m/s^2, h is the maximum height, and m is the mass of a blood particle, which cancels out. Simplifying this, we get an equation for the height of the blood fountain.

Plugging in numbers, the maximum height of this blood fountain is about 16 meters (or ~50 feet). In the video, the whale's innards spray sideways, so it doesn't reach nearly as high.

Now back to our original question. What was the pressure inside this whale? To figure this out, we need to use an idea known as Bernoulli's principle. It relates the pressure inside the whale (P_in) to the pressure outside the whale (P_out), the density of the blood (ρ), and the speed of the blood (v).

This equation will give us the pressure imbalance that builds up on the whale's body, due to all the gas that is building up.

The density of blood is 1025 kilograms/cubic meter, and the speed I know from above. The pressure outside the whale is just 1 atmosphere. Plugging in numbers, I find that the pressure inside the bloated sperm whale was about 2.6 atmospheres, and the pressure imbalance on the whale's body (the pressure inside minus the pressure outside) is therefore 1.6 atmospheres.

That sounds like a lot, but as it turns out, this is way less than the maximum pressure imbalance that a whale's body can withstand. Based on this analysis, I'd conclude that the whale wouldn't have spontaneously exploded for a while, if the biologist hadn't poked it open!

But, as always, I'm making a simplifying assumption. I've assumed that the whale is essentially a big balloon, with rubber replaced by whale flesh and blubber. (As the joke about physicists goes, assuming a spherical whale.) In reality, a whale has orifices, and so the gas will likely escape through the path of least resistance - which might be the mouth, the anus, or the genital track. (The more you know!) This might be why the pressure that builds up as a whale explodes is much less than the pressure it takes to tear whale flesh - the gases find the easiest way out.

So for future reference, if you find yourself standing next to a beached whale, stay away from the orifices! (In all seriousness, hopefully this video and calculation has convinced you that you shouldn't go anywhere near a bloated beached whale, unless you are Joy Reidenberg.)

Hat tip to Nadia Drake for asking this question on twitter and getting some great responses. Fun and nerdy conversations like these are the reason I love twitter.

Homework Problem: How much kinetic energy is released when a whale explodes? And how much energy would be released if you were to light a match inside it? (Here's one estimate. For the love of all that is holy, do not try this at home.)

Want to learn more? Inside Nature's Giants is a great (and gruesome) documentary where expert anatomists dissect a beached sperm whale, and describe some of the totally fascinating science behind this impressive mammal. Streams for free in the US and UK.

Update: Via twitter, John Hutchinson alerted me to this experimental biology paper, which estimates that the net pressure on the blubber of bloated whales is about 3 to 5 atmospheres (see Fig 6c). So the estimate in this post of 1.6 atmospheres is within a factor of 2 of these values, which is encouraging.