Chapter 3: Dynamics

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Imagine you're at the end of a long runway.

You know, you're looking out the window, the jet engines roar to life, and as the plane accelerates down the tarmac, you feel that that firm, undeniable push back into your seat.

Right.

It's a very specific feeling.

Exactly.

And that physical pressure, that is the invisible matrix of the universe at work.

So today we are decoding that exact sensation for you.

We're going deep into dynamics, which is basically the fundamental physics of how and why things move.

Yeah.

And to do that, we need to build a rock solid foundation, mapping out the precise rules that govern all motion.

We're using chapter three dynamics, explaining motion from the Cambridge International AS and A level physics course book.

Because we aren't just going to like memorize formulas to solve abstract math problems.

That's not the goal of the steep dive.

We are going to look under the hood of reality to understand why the physical world behaves the way it does.

Exactly.

From a skydiver and freefall to literally the structural integrity of the floor beneath your feet.

I mean, everything follows the same elegant logic.

So before we can explain complex high speed motion, like a jet plane taking off, we have to start with the most fundamental rule of dynamics,

which is Newton's second law of motion.

Right.

The big one.

Yeah.

And at its core, it boils down to an incredibly famous equation.

F equals ma,

resultant force equals mass times acceleration.

But how does that actually play out in real life?

Well, let's say we have a massive freight train driving through a tunnel.

That's an example straight from the text.

Okay.

How do we apply F equals ma to something that huge?

You start by looking at the forces pushing it forward.

If that train has a mass of 10 ,000 kilograms and the locomotive's motors generate a forward thrust, a resultant force basically of 20 ,000 Newtons.

Right.

We can predict exactly how its speed will change.

Acceleration is simply the resultant force divided by the mass.

So that's the 20 ,000 Newtons divided by the 10 ,000 kilograms.

Exactly.

Which gives the train an acceleration of two meters per second squared.

Meaning for every second that passes, the train speed increases by two meters per second.

You got it.

But wait, what happens when the train reaches the station?

Like it has to hit the brakes.

Does the math just break down when things slow down?

Not at all.

The mathematical relationship just flips its orientation.

When a vehicle decelerates, we treat that acceleration as a negative value.

Oh.

Yeah.

So if you know the train decelerates at three meters per second squared, you plug a negative three into the equation, you multiply the 10 ,000 kilogram mass by a negative three acceleration, and you get a resultant force of negative 30 ,000 Newtons.

I always used to get tripped up by negative numbers in physics, but that minus sign isn't just a random math quirk, is it?

No, not at all.

It's telling you a physical truth.

Yeah.

It means the braking force is pushing in the exact opposite direction that train is moving.

Got it.

And there's some other great worked examples in the text for this, like the cyclist.

Right.

A 60 kilogram cyclist on a 20 kilogram bike.

So a total mass of 80 kilograms.

Exactly.

If they apply 200 Newtons of driving force, you just divide 200 by 80, yielding an initial acceleration of 2 .5 meters per second squared.

Or the breaking car example.

A 500 kilogram car braking from 20 meters per second to a dead stop in 10 seconds.

Right.

So first you find the acceleration, which is the change in velocity over time.

Negative 20 divided by 10 gives you an acceleration of negative two meters per second squared.

And then multiply that by the 500 kilogram mass to get a braking force of negative 1 ,000 Newtons.

Okay.

Let's unpack this a bit for you, the listener, because this mathematical relationship highlights something profound.

It really does.

Acceleration is directly proportional to force, meaning if you push twice as hard, the object speeds up twice as fast, but it is inversely proportional to mass.

I like visualizing this with an extreme contrast.

Imagine you are standing in a parking lot, trying to push a tiny smart car versus trying to push a massive heavy SUV.

Yeah.

Good analogy.

Right.

If you push them both with the exact same amount of muscle, the tiny car is going to accelerate much more noticeably because it has less mass.

In physics, mass is fundamentally a measure of an object's inertia.

Inertia.

Right.

Inertia is a property that dictates how stubbornly an object resists any change to its motion.

The more massive the object, the harder it is to speed it up, slow it down, or change its direction.

So we know that forces cause acceleration, but what actually are these forces?

The text has this whole zoo of forces, table 3 .2.

When we draw diagrams to predict motion, we use arrows to represent all these specific pushes and pulls.

Broadly speaking, you have forces that act at a distance and forces that require physical contact.

Okay.

The most obvious action at a distance force is weight.

That is the gravitational pull of the earth acting on an object's mass.

In diagrams, we always draw it as an arrow pointing vertically downward from the object's center of gravity, straight toward the center of the planet.

And then you have the contact forces, which, like the name implies, only happen when things physically touch or interact.

Like friction, which is the force rubbing against motion when two surfaces slide past each other.

Exactly.

Or a car engine pushing backward on the road, so friction actually pushes the car forward.

Right.

Or drag, which is the fluid resistance you experience moving through liquids or gases.

We also have up thrust.

That's the upward push keeping objects afloat in liquids or gases, right?

Like boats or hot air balloons.

Spot on.

And tension, the pulling force transmitted through stretched strings, ropes, or springs.

I have to admit, I get a little confused when we talk about just standing perfectly still, because like when I stand on the floor, I know gravity is pulling me down, but I don't feel like a force is pushing back up on me.

It just feels like, well, the floor, how can an inanimate object push?

What's fascinating here is that this is a massive hurdle when you're first mastering dynamics.

We assume inanimate objects are completely rigid, but at the atomic level, they aren't.

They aren't.

No.

When you stand on the floor, your weight actually compresses it slightly.

You are physically pushing the atoms of the floor closer together than they naturally want to be.

Wait, seriously.

So the floor is essentially bending under me, even if I can't see it.

Yes.

And as those atoms are squished together, the interatomic forces act like billions of microscopic springs.

Oh, wow.

They naturally want to push back to their original position.

That collective upward push from all those tiny atomic springs is what we call the normal contact force, and it pushes back up against your weight, keeping you from falling through the floor.

That is genuinely mind -bending.

I'm just squishing trillions of atomic springs right now.

But speaking of weight, we need to look closer at gravity.

It's the one force we literally cannot escape.

We talked about F equals ma, but there's a specific version of that, isn't there?

There is.

Weight equals mass times the acceleration of freefall.

W equals m rho.

And that concept goes back to Isaac Newton, right?

Yeah.

He was confined to his rural home during a plague outbreak when he observed an apple fall from a tree.

That simple everyday occurrence kind of sparked his theory of gravity.

Right.

He realized the force pulling the apple down is the exact same fundamental force keeping the moon in orbit around the earth.

And near the earth,

that gravitational acceleration, that lowercase jush, is about 9 .81 meters per second squared.

Exactly.

Which brings up a really famous, though maybe mythical story about Galileo dropping cannonballs.

Right.

He supposedly dropped a large cannonball and a small cannonball off the leaning tower of Pisa at the same time to see which would hit the ground first.

Because everyone assumed the physics dictates that they hit the ground at the exact same time.

The acceleration of freefall is entirely independent of mass.

Wait, wait,

I have to step in and push back on behalf of our listener here because common sense tells us heavy things absolutely fall faster.

Okay, I hear you.

If I stand on a roof and drop a stone and a feather, the stone drops like a rock and the feather lazily floats down.

You can't tell me they accelerate at the same rate.

Well, your common sense is noticing a very real phenomenon, but you are attributing it to wrong force.

Okay, explain.

The difference in their fall isn't because gravity pulls the heavier object with a greater acceleration.

It's entirely due to our old friend drag,

air resistance.

So the air is physically getting in the way.

Yeah, the feather has a large surface area relative to its tiny, tiny weight.

So as it falls, the air molecules push back against it significantly.

Right.

The stone, on the other hand, has a lot of mass concentrated in a small area.

So it essentially punches right through the air resistance.

If you remove the air entirely, the playing field is completely leveled.

And they actually proved this on the moon.

The Apollo 15 astronauts performed this exact experiment.

You did, yeah.

Since the moon has virtually no atmosphere and therefore no air resistance, an astronaut dropped a geological hammer and a feather at the exact same time, and they fell side by side and hit the lunar dust at the precise same instant.

It's a beautiful demonstration of pure dynamics stripped to the variables of Earth's atmosphere.

But because we do live in an atmosphere, if gravity is always pulling us down and friction and air resistance are always slowing us down, it creates this massive everyday illusion about how motion works.

For centuries, people were completely confused by this.

Yeah, even going all the way back to the ancient Greek philosopher Aristotle, people observed things like, say, an elephant pulling a massive tree trunk through a

person peddling a bicycle.

Right.

In all these everyday cases,

the moment the pulling or pushing force stops, the motion stops, the tree stops dragging the bike coast to a halt.

So the logical, albeit totally incorrect conclusion was that a moving object needs a continuous force to keep it moving.

Which is fundamental misconception.

Aristotle and his followers just couldn't see the invisible thief constantly stealing their momentum, which is friction.

Here's where it gets really interesting for you guys.

In the 17th century, astronomers started looking through telescopes and tracking the planets.

And they saw these massive celestial bodies moving freely through the vacuum of space.

There were no giant elephants pulling them, no engines pushing them.

They just kept moving.

And that realization shattered the old worldview and gave birth to Newton's first law of motion.

Which says what exactly?

It states that an object at rest will stay at rest and a moving object will continue moving at a steady speed in a straight line unless a resultant force acts on it.

So if I were to ride a bicycle in a magical, frictionless vacuum, I could pedal once to get up to speed and then I would just glide forever without ever having to pedal again.

You would glide for eternity.

That steady speed in a straight line is what we call uniform motion or constant velocity.

You only need a forward force to overcome friction and drag.

If those don't exist, motion is free.

This loops perfectly back to inertia.

Let's think about a massively overloaded supermarket trolley.

Oh yeah, a classic example.

Right.

It is so hard to get moving, it's hard to turn down the serial aisle, and it's incredibly hard to stop before you crash into the checkout counter.

Because that large mass gives the trolley a high inertia.

It violently resists any change to its state of uniform motion.

If it's still, it wants to stay still.

If it's moving, it wants to keep moving.

But, as we established, we don't live in a magical vacuum.

We live in a world filled with fluids, like air and water.

Which means we constantly have to calculate what happens when forward motion goes to war with drag.

And this introduces the concept of balanced versus unbalanced forces.

Okay, break that down for us.

Well, if the resultant force on an object is zero zero, meaning all the individual forces pushing and pulling on it perfectly cancel each other out, the object's velocity remains constant.

It doesn't speed up, and it doesn't slow down.

Let's visualize this with the textbook's example of a skydiver jumping out of an airplane.

The moment they step out a door, gravity pulls them down.

At that specific moment, gravity is an unbalanced force, so they accelerate downwards.

Right.

But as their speed increases, they are smashing into more and more air molecules every single second.

Like pushing your hand out the window of a moving car.

Exactly.

As they hit more air molecules,

the air resistance, the drag force pushing back up against them increases.

Eventually, they fall so incredibly fast that the upward drag force exactly equals their downward weight.

It's like a break -even point in a business.

The income of gravity is perfectly matched by the expenses of air resistance.

The forces are now balanced.

The resultant force is zero.

And therefore, their acceleration becomes zero.

They stop speeding up and begin falling at a constant maximum speed.

We call this their terminal velocity.

And when they finally open their parachute, they massively increase their surface area.

Right, which instantly spikes the air resistance, creating a massive unbalanced force pointing upwards.

That forcefully decelerates them until they reach a new, much slower, and much safer terminal velocity for This phenomenon also explains why tiny insects, or even mice, can survive falls from incredibly high buildings.

Yeah.

Their weight is very small, but their surface area relative to their weight is quite high.

So the drag force balances out their weight almost immediately.

They hit their terminal velocity very quickly.

And that speed is so low that they land completely uninjured.

It's amazing.

And we can map out these battles of forces mathematically.

Like the book has this worked example about a 500 kilogram car with a 300 newton forward force and 200 newtons of air resistance.

Right.

So the resultant force is just 300 minus 200, 100 newtons.

And then acceleration is force over mass.

So 100 divided by 500 is 0 .2 meters per second squared.

But you can also use this to find absolute top speeds, right?

Like treating the formula like a puzzle to figure out sports cars top speed.

Exactly.

Say the engine can only produce a maximum forward force of 500 newtons.

And as the car speeds up, the air resistance pushes back harder and harder.

The text gives us the formula for this specific car's air resistance.

F equals 0 .2 v squared.

Right.

The fascinating thing is how that air resistance scales.

It scales with the square of your velocity.

So to find the top speed, you set the max forward force equal to the drag.

So 500 equals B0 .2 v squared.

You divide 500 by B0 .2 to get 2500.

And the square root of 2500 is 50.

Yep.

50 meters per second, or roughly 180 kilometers per hour.

The squared part is the real kicker, isn't it?

That is the ultimate speed limit.

It absolutely is.

If your drag scales with your velocity squared, it means if you want to drive twice as fast, you don't just face twice the air resistance.

Two squared is four.

You face four times the air resistance.

So if you want to three times as fast, you face nine times the resistance.

That explains why hyper cars need these absurdly massive engines just to squeeze out a few extra miles per hour at their top end.

The drag wall becomes nearly impossible to push through.

And athletes actively manipulate this physics every day.

Racing cyclists wear teardrop -shaped aerodynamic helmets and skin -tight clothing to minimize their surface area and drastically lower that drag equation so they can hit higher top speeds.

Oh, and conversely, sprinters will sometimes attach actual parachutes to their waist during resistance training.

Exactly.

They intentionally increase their drag surface area to make their muscles work harder against those unbalanced forces.

The mechanics of drag are so fascinating.

Up until now, though, we've basically mastered how forces act on a single object.

A single skydiver falling, a single car pushing against the air.

But what Ah, this brings us to Newton's third law of motion.

Okay, lay it on us.

The third law states that when two bodies interact, the forces they exert on each other are equal in magnitude and opposite in direction.

Now, I often hear this summarized as, for every action, there is an equal and opposite reaction.

But the physics community really warns against using that phrasing, right?

They do, yeah.

Using the words action and reaction creates a dangerous misconception.

It implies a sequence of events.

Like a domino effect.

Right.

It implies that I push you and then a moment later you push me back as a reaction.

That is fundamentally wrong.

These forces happen entirely simultaneously.

One doesn't cause the other sequentially.

They are born at the exact same instant.

And they have two very specific criteria they always follow.

Right.

They always act on different bodies and they're always the exact same type of force.

Let me try to make this with a physical example.

Let's say I'm walking and I accidentally step down hard on someone's toe.

How does the third law apply there?

It's a perfect illustration.

At the exact moment your foot exerts a downward contact force on their toe, their toe exerts an equal and opposite upward contact force on your foot.

Even though my foot is doing the stepping.

It doesn't matter who initiated the movement.

The interaction is a two -way street.

It's the same type of force.

Right.

A contact force and it acts on two different bodies.

Your foot and their toe.

Equal in magnitude, opposite in direction, occurring simultaneously.

This applies to so many things when you think about it.

If you are swimming, you push the water backward with your hands.

That's a contact force.

Simultaneously, the water pushes your hands forward with the exact same amount of force.

Yep.

That's why you propel through the pool.

Or a rocket in space.

People often mistakenly think a rocket pushes against the air to move, but there's no air in space.

The rocket's engine pushes the exhaust gas out the back and simultaneously the exhaust gas pushes the rocket forward.

Same exact principle.

That is a stellar application of the law.

Okay.

We have covered an immense amount of ground.

Newton's laws, the different types of forces, gravity, terminal velocity, and inertia.

But to wrap this deep dive up, we need to talk about the language we use to share all these discoveries.

Right.

The units.

Yeah.

If scientists around the globe are going to share calculations about motion, we need a universally agreed upon language.

We need the International System of Units, or the SI system.

It is completely non -negotiable for modern engineering and science.

Think about a complex manufacturing supply chain.

Imagine an engineering company in Taiwan manufacturing a highly precise engine part for a car that is going to be assembled in a factory in India.

If the engineers in Taiwan and the engineers in India have even a microscopic fraction of a hair disagreement on exactly how long a millimeter is or how heavy a kilogram is, that engine simply will not fit together.

It will fail.

Exactly.

To prevent that chaos, we use SI base units.

These are the foundational building blocks of all physical measurement.

So what are they?

You have meters for measuring length, kilograms for mass, seconds for time, amperes for electric current, Kelvin for thermodynamic temperature, and moles for the amount of substance.

And every other unit in physics is basically a mashup, a derived unit built by combining those core base units.

Take the Newton, for example, the unit of force we've been talking about all day.

How is that derived?

We define it using the very first equation we discussed.

F equals ma,

force equals mass times acceleration.

Mass is measured in kilograms.

Acceleration is measured in meters per second squared.

Therefore, one single Newton is by definition exactly equal to one kilogram meter per second squared.

There's also a really important practical rule when dealing with prefixes like kilo, milli, or micro, especially for students.

If you are doing calculations and you square or cube a unit, you absolutely have to square or cube the prefix too.

Oh, this trips up so many students.

One cubic centimeter is not just 10 to the negative two meters cubed.

It is 10 to the negative two all cubed, which actually makes it 10 to the negative six cubic meters.

It's a massive difference mathematically.

Massive.

So bringing all this math and measurement together with my favorite question.

So what does this all mean?

Like, why should you care so deeply about base units and something physicists call homogenous equations?

Because homogenous equations are the ultimate physics cheat code.

A homogenous equation simply means that the base units on the left side of the equal sign perfectly match the base units on the right side.

So it's like a built -in testing mechanism.

Yes.

If you were sitting in a high pressure exam or trying to solve a complex engineering problem and you panic and forget if you wrote a formula down correctly, you don't have to guess.

You just break both sides down into their fundamental base units.

Kilograms, meters, seconds.

Oh, I see.

If the units on the left perfectly match the units on the right, the physics is sound.

If they don't match, you instantly know you've made a mistake.

It is a built -in lie detector for your math.

A lie detector for your math.

I absolutely love that.

Before we go, I want to leave you with a final thought to mull over.

We've talked a lot about equations, diagrams, and historical figures today, but remember that you aren't just studying physics on a page.

You are constantly living it.

It's everywhere.

Every single time you move your arm, every time you accelerate your car onto a highway, every time you accidentally drop your keys, you are executing perfect, elegant math.

You are living constantly inside an invisible matrix of balanced and unbalanced forces.

Yeah.

The next time you take a simple step down the sidewalk, think about it.

The earth is pushing back up on you with the exact same force you are pushing down on it.

That is an amazing image.

The invisible math of every single step.

Thank you so much for sitting down and studying with us today.

On behalf of the last minute lecture team, we hope this deep dive helped clear the muddy waters of dynamics.

So the next time you are sitting in that passenger plane, feeling that firm push back into your seat, you'll know exactly what the physical world is doing.

Keep questioning, keep learning, and we will catch you on the next deep dive.