## Wednesday, October 23, 2013

### Tip o' the Lance

My weekend activities have provided me with ample blogging fodder of late. This past weekend I went to a local Renaissance Festival and, among other things, watched some real life jousting. That is, actual people got on actual horses and actually rammed lances into each other, sometimes with spectacular results.

 I didn't take this picture. It's from the Renn Fest website. I just think my blog needs more visuals.
At one point a lance tip broke off on someone’s armor and went flying about 50 feet into the air. A friend wondered aloud what kind of force it would take to achieve that result, and here I am to do the math. This involves some physics from last year as well as much more complicated physics that I can’t do. You see, if a horse glided along the ground without intrinsic motive power, and were spherical, and of uniform density… but alas, horses are not cows.

Anywho, as to the flying lance tip, the physics is pretty easy. Now, I can’t say what force was acting on the lance. The difficulty is that, from a physics standpoint, the impact between the lance and the armor imparted momentum into the lance tip. Newton’s second law (in differential form) tells us that force is equal to the change in momentum over time. Thus, in order to calculate the force of the impact, I have to know how long the impact took. I could say it was a split second or an instant, but I’m looking for a little more precision than that.

Instead, however, I can tell you how much energy the lance tip had. It takes a certain amount of kinetic energy to fly 50 feet into the air. We’re gonna say the lance tip weighs 1 kg (probably an overestimate) and that it climbed 15 meters before falling down. In that case, our formula is e = mgh, where g is 9.8 m/s2 of gravitational acceleration, and we’re at about 150 joules of energy. This is roughly as much energy as a rifle bullet just exiting the muzzle. It also means the lance tip had an initial speed of about 17 m/s. I’m ignoring here, because I don’t have enough data, that the lance tip spun through the air—adding rotational energy to the mix—and that there was a sharp crack from the lance breaking—adding energy from sound.

But this doesn’t conclude our analysis. For starters, where did the 150 joules of energy come from? And is that all the energy of the impact? Let’s answer the second question first. Another pretty spectacular result of the jousting we witnessed was that one rider was unhorsed. We can model being unhorsed as moving at a certain speed and then having your speed brought to 0. Some googling tells me that a good estimate for the galloping speed of a horse is 10 m/s.

So the question is, how much work does it take to unhorse a knight? With armor, a knight probably weighs 100 kg. Traveling at 10 m/s, our kinetic energy formula tells us this knight possesses 5000 joules of energy, which means the impact must deliver 5000 joules of energy to stop the knight. This means there’s certainly enough energy to send a lance tip flying, and it also means that not all of the energy goes into the lance tip.

We can apply the same kinetic energy formula to our two horses, which each weigh about 1000 kg, and see that there’s something like 100 kj of energy between the two. Not all of that goes into the impact, however, because both horses keep going. This is where the horses not being idealized points hurts the analysis. Were that so, we might be able to tell how much energy is “absorbed” by the armor and lance.

There is one final piece of data we can look at. I estimate the list was 50 meters long. The knights met at the middle and, if they timed things properly, reached their maximum speeds at that point. Let’s also say that horses are mathematically simple and accelerate at a constant rate. One of the 4 basic kinematic equations tells us that vf2 = vi2 + 2ad. So this is 100 = 0 + 2*a*25, and solving for a gets us an acceleration of 2 m/s2. Newton’s second law, f=ma, means each horse was applying 2000 newtons of force to accelerate at that rate. 2000 N across 25 meters is 50,000 joules of work. It takes 5 seconds to accelerate to 10 m/s at 2 m/s2, so 50,000 joules / 5 seconds = 10,000 watts of power. What’s 10,000 watts? Well, let’s convert that to a more recognizable unit of measure. 10 kW comes out to about 13 horsepower, which is about 13 times as much power as a horse is supposed to have. Methinks James Watt has some explaining to do.

One other thought occurred to me during this analysis. Some googling tells me there are roughly 60 million horses in the world. If a horse can pump out 10 kW of energy, then we have roughly 600 GW of energy available from horses alone. Wikipedia says our average power consumption is 15 TW, which means the world’s horses running on treadmills could provide 4% of the energy requirements of the modern world. This isn’t strictly speaking true, because there will be losses due to entropy (and you can’t run a horse nonstop), but it’s in the right ballpark. Moral of the story? Don’t let anyone tell you that energy is scarce. The problem isn’t that there isn’t enough energy in the world; it’s that we don’t have the industry and infrastructure necessary to use all the energy at our disposal.