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[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."