(electronic tones) - Energy is a powerful tool for predicting the behavior of our universe from quantum to cosmological scales.

It's also pretty good for siege warfare.

Let's see how.

(funky electronic melody) In a recent episode we talked about one of the most powerful and misunderstood concepts in all of physics.

We asked, what is energy?

The term has been somewhat hijacked by new ageism.

That's cool, physics steals words all the time.

Unfortunately, the metaphysical use has become a rather vague catch-all for any intangible influence.

In physics, energy is still intangible.

It's not a thing, it's a property that things can have.

But the energy of physics is anything but vague.

It's a very clearly defined in a way that makes it an incredibly powerful tool for calculation.

Today, we're going to use that tool.

In fact, we're going to calculate the very tangible effects of this intangible abstract stuff.

Before we get started, you should pause here and watch the previous episode if you haven't already.

Good?

Okay, let's do one more quick review.

Any conservative force that changes an object's speed over some distance will change that object's speed in exactly the opposite way on the reverse path.

More, it doesn't actually matter what path the object takes between two points under the influence of that force.

The change in speed for a given object will be the same as long as the start and end points are the same.

Of course, that change in speed will be different for objects of different masses.

After all, heavy things are harder to accelerate.

But Newton's Laws tell us that when two objects with different masses are accelerated by the same force over the same path, the quantity half mass times velocity squared has to be constant.

We name that quantity kinetic energy.

If we knew the exact amount of kinetic energy that would be gained or lost by traveling between two points, then we can keep track of the potential for future gains or losses of motion.

We do this by defining this thing called potential energy.

To find the right way, the sum of kinetic and potential energy, or motion and potential for motion remains constant.

And not just for one particle, but for any system of any number of interacting particles.

After all, the interactions between particles are ultimately due to fundamental forces which are always conservative.

Kinetic and potential energy are defined as combinations of more basic quantities.

For example, position, velocity, and mass, and these combinations are chosen so that their sum is conserved.

But it's actually remarkable that there's any such combination of quantities that is conserved.

This fact gives us insight into the fundamental symmetries of nature.

There's something we'll get back to, but today I wanna highlight the power of energy as a tool in calculation.

We could get into fluid, or stellar dynamics, or even quantum mechanics, but no, we're gonna talk about something much cooler: the trebuchet.

If you're not familiar, shame on you, but here's the deal.

The trebuchet is, I suppose, a type of catapult.

But it's so much more.

The internet agrees that the trebuchet is the greatest of all medieval siege engines.

Opinion is still divided on whether Chuck Norris would win a fight against a trebuchet.

The hyperbole is kind of warranted.

The trebuchet is incredibly efficient at converting the potential energy of a massive counter-weight into the cattle-destroying kinetic energy of a projectile.

It relies on the mighty power of the lever.

As Archimedes once said, "Give me a lever and a place to stand, "and I will move the Earth, "or hurl a 90 kilogram stone over 300 meters."

Let's see how this works.

The trebuchet's counterweight pulls down a short lever arm which pivots the longer arm upwards.

The counterweight travels through a short arc while the end of the opposing arm travels through a much longer arc in the same amount of time.

In order to do that, the tip of the long arm must reach a very high speed.

A sling containing a projectile is attached to that tip.

The sling rotates through a much larger angle than the arm, further increasing the speed at which the projectile is released.

Okay, so you're a savvy warlord, and you appreciate the awesomeness of the trebuchet.

One day you're laying siege to your enemy's fortress, as you do, you wanna figure out the speed of impact of a trebuchet projectile based on the mass and the movement of the counterweight.

Now, it's possible to figure out the equations of motion of a trebuchet in terms of Newton's Laws, with a complicated series of force factors, some gnarly geometry, and of course, some calculus.

But there's no time for all of that.

You're a busy warlord with enemies to vanquish.

This is where energy comes in.

The law of conservation of energy tells us that the sum of kinetic and potential energies of the projectile and the counterweight are conserved.

You should be able to use simple energy arguments to calculate for your enemies' destruction.

I should add that we're making a few assumptions here.

We' are assuming that you've built the perfect trebuchet.

No energy is transferred to the structure of the trebuchet through friction or other motion.

All energy stays in the projectile and the counterweight.

Also, there's no air resistance.

These assumptions aren't entirely reasonable.

But in a real energy calculation, losses due to non-conservative effects like friction and air resistance can be accounted for.

We'll also assume that the mass of the lever arm is tiny compared to the mass of the counterweight and the projectile.

Now, the mass of the lever arm could be included by talking about the change in height of the center of mass of the whole counterweight arm system, but for today we'll just talk about the change in height of the counterweight.

Our first question requires no math.

You fly your trebuchet, and the projectile flies upwards to slam into the top of the tall fortress wall.

Nice one.

Now you try a different shot.

You raise the counterweight to the same height as last time and let it fall.

This time, you release the projectile a little earlier so it takes a more vertical trajectory.

It flies high into the air and then falls to hit exactly the same spot on the wall.

You notice that in both of your shots the counterweight continued to swing after release.

And in both of your shots, that post-release swing reaches the exact same height.

So, my question is this: which of your two shots does the most damage?

Assume that damage depends only on the speed that the projectile hits the wall.

Okay, now I have an extra credit question.

This time, we'll use numbers.

Let's say you drop a five ton counterweight from a height of eight meters above the ground.

All that good lever action swings a 90 kilogram stone from ground-level, releasing it at some point in the upward arc.

After releasing the stone, the trebuchet arm continues to swing.

The counterweight swings to a lowest point, one meter above the ground, and continues its arc, ultimately rising to a height of two meters before swinging back again.

Meanwhile, the stone travels 300 meters to strike the fortress wall 15 meters above the ground.

My question, how fast is the stone traveling at the moment of impact?

The really surprising thing about this problem is that you don't need to know the lengths of the arms, the release point of the ball, or any of that.

It's enough to know the start and end locations of the counterweight and projectile, along with their masses.

That's the power of energy.

To enter this challenge, write up your answers to either question, you don't have to do both.

Write your answer neatly, explain your reasoning, and show your work.

Draw diagrams if you need to.

Submit your answers within two weeks of release of this episode to PBSSpaceTime@gmail.com.

Use the subject line "Trebuchet Challenge."

Use exactly that subject line, and check your spelling, because we sort by subject line.

We'll choose six correct entries to receive Space Time t-shirts.

That way next time you besiege a fortress, you can do it under the banner of Space Time.