[JINGLE PLAYING]

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.

[ELECTRONIC MUSIC]

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 Age-ism.

That's cool.

Physics steals words all the time.

Unfortunately, the metaphysical use

has become a rather vague catchall

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 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?

OK, 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 named that quantity kinetic energy.

If we know the exact amount of kinetic energy

that will 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.

Defined 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.

Now, 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.

And it's something we'll get back to.

But today, I want to 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 going to 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 counterweight

into the castle-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 counter-weight 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.

OK, 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 want to 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

vectors, some gnarly geometry, and of course, some calculus.

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 enemy's destruction.

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

We're 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 fire 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.

OK, 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 1

meter above the ground and continues

its arc, ultimately rising to a height of 2 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, explaining 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."