Chapter 3: Motion in Two or Three Dimensions

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Buckle up everybody because today we are going deep into the world of motion.

Yeah, motion.

But not just like your typical straight line stuff.

Exactly.

We're talking about curves and arcs and all the cool ways things move when there's more than one dimension involved.

Absolutely.

And, you know, we've got this physics textbook with all the nitty gritty details.

So our goal today is to unpack that and make it make sense.

Kind of like a translator for the laws of physics.

Exactly.

You know, break it down so we can actually understand how the world works.

Right.

And to get us started, let's dive right into a little brain teaser from the

Okay.

Speedometers locked in.

Yep.

Totally constant speed.

Now the question is, are they actually accelerating?

Hmm.

So I'm guessing there's a trick here.

You bet there is because if you only think about acceleration as speeding up or slowing down.

Yeah.

Like in a street line.

Right.

Then it seems like the answer should be no.

Yeah.

No change in speed.

No acceleration.

Yeah.

But in physics,

acceleration is a bit more nuanced.

It's about any change in velocity.

And velocity is a bit more than just speed.

It is.

Velocity includes both how fast something's going and the direction it's moving.

So if the direction is changing, even if the speed's the same.

You've got it.

That cyclist is experiencing acceleration.

And that idea is super important for understanding all sorts of motion.

It's a game changer.

Totally.

So whether it's that curveball and baseball.

Yeah.

Or how satellites stay in orbit around earth or even how you perceive the motion of other cars when you're driving.

It's all connected to these fundamental ideas about motion in more than one dimension.

Right.

So how are we going to break all this down today?

Well, we'll start with the building blocks vectors.

Those are our tools for describing position and velocity.

Okay.

So kind of like coordinates on a map, but with more information.

Exactly.

And then we'll tackle that acceleration along curves head on.

That'll definitely answer our cyclist question.

Definitely.

From there, it's projectile motion.

Think like a baseball flying through the air and then we'll zoom in on circular motion, both uniform and non -uniform.

Yeah.

So constant speed versus changing speed.

Exactly.

And to wrap things up, we'll switch perspectives and look at how motion appears from different moving viewpoints.

Relative velocity.

That's always a fun one.

It's all about who's watching.

So shall we get started with these vectors?

Let's do it.

Awesome.

Because I know in everyday life, if I want to tell you where something is, I might say it's over there or it's three steps to the left.

Right.

But when we're talking about more than one dimension, like up, down, left, right, forward, backward, just saying a distance isn't enough.

We need a way to say exactly which direction too.

And that's where vectors come in handy.

Okay.

So vectors are like giving someone directions, but way more precise.

They're like a set of very specific instructions go this far in that direction.

Right.

And our trusty textbook uses RS to represent the position vector.

Right.

So if you imagine a 3d space, like a room, the X, Y, and Z axis,

like length, width, and height.

Exactly.

We can pinpoint any location with those three coordinates.

So that's what the position vector does.

It tells us exactly where something is in that space.

So mathematically, we write it like this RSD plus E plus ZK.

Okay.

So X, Y, and Z are our coordinates and those D, and K things.

Those are like our labels for the X, Y, and Z directions.

Okay.

Got it.

So what about if our object moves from one point to another?

Well, then we're talking about displacement error S, which is just the arrow.

Like literally an arrow.

Yeah.

Pointing from the starting point to the ending point.

Okay.

So that arrow tells us both how far the object moved and in which direction.

Precisely.

And we can define an average velocity, but now it's the average velocity vector VSAF.

Okay.

So it's not just speed anymore.

It's speed and direction.

Right.

And it tells us the overall rate and direction of that displacement over a certain time interval.

And that's written as VSAF, A is ER.

Exactly.

And what's really cool is that each part of this vector, it's X, Y, and Z components.

Kind of like breaking it down into smaller pieces.

Exactly.

Each one behaves just like the average velocity we're used to in one -dimensional motion.

So like the X component is just the change in X divided by the change in time.

Exactly.

Okay.

That makes sense.

Yeah.

That's average velocity, but what if we want to know the velocity at a very specific moment?

Then we need instantaneous velocity VS.

So it's like the difference between knowing your average speed for a whole road trip.

Right.

Versus looking at your speedometer at one exact moment.

Exactly.

It tells you precisely how fast and in what direction the object is moving right then and there.

Okay.

And how do we get that instantaneous velocity?

Mathematically, it's defined as the derivative of the position vector with respect to time.

Okay.

That sounds a little complicated.

It sounds fancy, but it's just a way of capturing that instantaneous rate of change.

Okay.

And this instantaneous velocity vector is always tangent to the path the object is taking.

Yeah.

It's like if the object suddenly decided to go in a straight line, that's the direction it would head in.

I like that analogy.

It's like the roller coaster at any point along the track,

the instantaneous velocity is pointing in the direction of the track at that exact spot.

Perfect analogy.

Okay.

But how do we actually calculate VS?

Well, we can break it down into its components just like with the position vector.

So VXV and VZ.

Exactly.

Each one is the rate at which that coordinate is changing with time.

And we can get those by taking those derivatives.

Exactly.

And if we put those components together, we get the full instantaneous velocity vector.

VS is VXDN plus VN plus VZKN.

Yep.

And the overall speed.

That's just the magnitude of this velocity vector.

So how fast it's going regardless of the direction.

Right.

And we find that using a sort of 3D Pythagorean theorem,

VXO off VXS plus VZXO off.

And if we're just looking at motion in a flat plane like the Xi plane.

We can even find the direction of this velocity using trigonometry.

Tan ESS VQSX.

Or 1A is the angle relative to the X axis.

Exactly.

Our textbook even gives that cool example with the Mars Rover moving on the Martian surface.

Oh yeah.

They give us equations that describe how its X and Y coordinates change with time.

Right.

So we can track its position over time.

Yeah.

And then they use those equations to find its exact location and velocity at a specific time.

Yeah.

And that really connects the abstract math to the actual movement.

It does.

And they make a good point about how the average velocity over a certain period.

Isn't necessarily the same as the instantaneous velocity at the end of that period.

Right.

So the Rover's overall journey from point A to point B might be one thing, but its specific speed and direction at a particular moment could be totally different.

Exactly.

Okay.

So now let's talk about acceleration, Asis.

All right.

Let's dive in.

Because I know most people think of acceleration as just speeding up.

Yeah.

Like stepping on the gas pedal.

But in physics, it's way more than that.

It's much broader.

It's any change in velocity.

And since velocity has both magnitude and direction,

acceleration can happen in a few different ways.

Right.

So an object can speed up, it can slow down, or it can just change direction.

And often it's a combination of all three.

Exactly.

And just like velocity, we have average acceleration, ASAF, and QLS.

Right.

That's the overall change in velocity over a certain time interval.

And then we have instantaneous acceleration, S equals DVSDT.

Which is that change in velocity over a super tiny time interval.

So it's that acceleration at one precise moment.

Exactly.

And that direction of the acceleration vector is super important, especially when we're dealing with curved paths.

Absolutely.

For an object on a curved path, that instantaneous acceleration vector always points towards the inside of the curve.

It's like the force that's constantly nudging the object to change direction.

Right.

And that brings us back to our cyclist question.

Oh, yeah.

So even if the cyclist is keeping their speed perfectly constant,

the fact that they're going around a curve means their direction is constantly changing.

And if the direction's changing, that means their velocity is changing.

And change in velocity equals acceleration.

Bingo.

Even at a constant speed, they're accelerating.

The textbook even uses the example of being a passenger in a car that takes a sharp turn.

Oh, yeah.

You feel like you're being thrown outwards.

Exactly.

Yeah.

But what's really happening is the car is accelerating inwards to change its direction.

Your body just wants to keep going straight.

Right.

Inertia.

Yeah.

It's like your body's resisting that change in direction.

Okay.

So constant speed, but changing direction equals acceleration.

Got it.

Nailed it.

Now, how do we handle this acceleration mathematically?

Same as before, we can look at its components, x, a, and as.

Each one tells us how that part of velocity is changing.

Right.

So x, for example, is the rate of change of the x component of velocity dvx dt.

Or if we go back to position, it's the second derivative of the x coordinate with respect to time dxt.

Exactly.

And we have similar expressions for a and as.

And our Marth Rover example shows how we can calculate both the average acceleration over an interval and the instantaneous acceleration at a specific moment.

It's a nice concrete example.

It is.

And the textbook also talks about breaking down acceleration into components that are parallel or perpendicular to the velocity vector.

Ah, yes.

That's a really insightful way to understand what acceleration is doing to the object's motion.

So it's like acceleration is wearing two hands.

Exactly.

The part of acceleration that's parallel to the velocity, we call that a.

That's the one in charge of changing the speed.

Right.

If it points in the same direction as the object speeds up, if it points opposite, the object slows down.

Makes sense.

And the part that's perpendicular to the velocity that's ash tag.

And that one's responsible for changing the direction of the object's motion.

So that's what makes a curve.

Exactly.

So if the acceleration is purely parallel to the velocity.

The object will move in a straight line, but its speed will change.

Right.

And if the acceleration is purely perpendicular.

Then the object will move along a constant.

And if we have a bit of both.

Then we'll get a curved path and a changing speed.

So it's all about how those two components work together.

Absolutely.

And the textbook gives this great example of a skier on a ski drum.

Oh yeah, I remember that one.

As they go down the straight part of the ramp, their acceleration is mostly parallel, making them speed up.

And then when they hit the curved part.

A big chunk of their acceleration becomes perpendicular, forcing them into that curved trajectory.

And when they land and start to slow down.

There's a component of acceleration acting opposite to their velocity.

I like how they even revisit our Mars rover example and calculate those parallel and perpendicular components.

Yeah, at that T2 .0 seconds mark.

Right.

And they find that the rover is speeding up and turning at that instant.

They also talk about a sled going over the top of a hill.

At that exact moment at the very peak.

Yeah, the acceleration due to gravity is straight down, which is perpendicular to the sled's velocity.

So at that moment, the acceleration is purely changing its direction.

It's a good illustration of how those components can work.

It is.

So we've got position velocity and acceleration as vectors down.

Check, check and check.

Now let's talk about something we see all the time.

Projectile motion.

Ah, projectiles.

Those are fun.

Totally.

So anything that's thrown or launched into the air.

Like a baseball, a football, a frisbee.

Exactly.

Those are all projectiles.

What makes them so special?

Once a projectile is in the air, the only force acting on it, ignoring air resistance, of course.

Yeah, we're keeping things simplified here.

Right.

Is gravity.

That means its path, its trajectory is determined by its initial velocity.

So how fast and in what direction it was launched.

Exactly.

And by that constant downward pull of gravity.

And importantly, all the action happens in a single vertical plane.

Yeah, no weird corkscrews or anything.

Okay.

So the big idea here is that the horizontal and vertical movements of a projectile are totally independent.

Yeah, that one can be a bit mind bending.

But think about this.

If you drop a ball straight down and at the exact same time you launch another ball horizontally.

From the same height.

Both balls will hit the ground at the exact same time.

It's true.

The fact that one ball is moving sideways has no effect on how quickly it falls vertically.

Right.

Gravity treats them both equally.

They both experience that same downward acceleration.

So how do we translate this into equations?

Well, since there's no horizontal force ignoring air resistance.

Right.

The horizontal acceleration acts as zero.

And the vertical acceleration.

Yeah, that's just orgy the acceleration due to gravity negative because we're defining upwards as positive.

Okay, so constant acceleration in both directions,

which means.

You can use those same equations of motion we learned for one dimensional motion, but we apply them separately to the horizontal and vertical directions.

So we end up with two sets of equations, one for x and vx and one for y and vi all as functions of time.

Exactly.

And those equations depend on the initial position and then initial components of the velocity v zero x and v zero.

So that's the launch speed v zero and the launch angle zero.

Right.

And we can use some trig to break that initial velocity into its horizontal and vertical components.

So v zero x equals v zero zero cosine zero and v two equals v zero sine zero.

Yep.

And if we plug those into our equations, we can actually map out the entire path of the projectile.

And that path that turns out to be a parabola.

Y se zero x d phi zero x four phi zero x zero.

That's the equation.

Of course, in the real world, air resistance messes things up a bit.

It does makes the path not quite a perfect parabola.

And it reduces the range and the maximum height.

Yeah, like with a baseball, for example.

Right.

But even if something gets a really complex push at the start.

Like a skier flying off a ramp.

Once they're airborne, it all boils down to those simple equations.

Yep.

x equals zero and a equals g.

Their initial speed and angle set them on their path.

But from then on, it's all gravity.

So what's the best way to tackle projectile motion problems?

The textbook gives some good advice.

Figure out what you know and what you need to find.

Right.

The knowns and unknowns.

Then set up a clear coordinate system so you know which way is which.

And then translate the words of the problem into those mathematical conditions.

Right.

So if they say highest point,

you know, that means the vertical velocity v must be zero at that moment.

They go through a couple of examples, a motorcycle jumping off a cliff and a batted baseball.

Yeah, those are good illustrations.

They show how we can use these equations to calculate all sorts of things.

Position, velocity, time of flight, maximum height, horizontal range, you name it.

And they even give us those formulas for the maximum height h and the range r.

When the projectile is launched and lands at the same level.

But those formulas only apply in that specific scenario.

Right.

If you launch from a window and land at a different height, you can't just plug and chug.

And that monkey and dart gun demonstration.

That's a classic.

If you aim directly at a monkey that drops the instant you fire, you'll always hit it.

It's counterintuitive, but it makes sense when you think about the independence of those horizontal and vertical motions.

Right.

Gravity affects both the dart and the monkey in the same way.

It's a great example of how these principles play out in unexpected ways.

Okay.

We've covered projectiles extensively.

Projectiles mastered.

Now let's move on to motion.

In a circle, we've touched on it with our accelerating cyclist.

Right.

Time to go full circle.

So uniform circular motion that's moving in a perfect circle at a constant speed.

Exactly.

So the object is tracing out a circle, but its speed stays the same.

But even though the speed is constant, the velocity is not.

Because the direction of motion is always changing.

And change in velocity means acceleration.

And in uniform circular motion, this acceleration always points toward the center of the circle.

It's always perpendicular to the velocity.

And we call that centripetal acceleration.

Center seeking.

And there's a formula for it, right?

Yeah.

A rod is, well, V varch R, where V is the constant speed and R is the radius of the circle.

So the faster the object is moving or the smaller the circle.

The greater the centripetal acceleration.

Makes sense.

The geometry and vectors.

It's a cool derivation.

It is.

It shows how the change in velocity is tied to the instantaneous velocity and the shape of the circle.

And they also give us a way to express our rod using the period of the motion T.

So the time it takes to complete one full circle.

Right.

Which leads us to our rod and two four tar T.

They use this in the carnival ride example.

Yeah.

They calculate the centripetal acceleration based on the radius of the ride and the time it takes to go around once.

And then there's that sports car on a curve road example.

By knowing the car's speed and the maximum lateral acceleration it can handle.

Which is basically the centripetal acceleration.

Right.

They figure out the smallest radius the car can safely turn at.

So a tighter turn at the same speed needs a much bigger centripetal acceleration.

Makes sense.

That's why you have to slow down for sharp turns.

Now what about if the speed isn't constant while something is moving in a circle?

Ah, then we've got non -uniform circular motion.

Where the speed changes as it goes around.

Exactly.

We still have that radial component of acceleration R equals VOR.

But its magnitude is changing because the speed is changing.

Right.

And we also introduce a new component.

Tangential acceleration.

Bingo.

That's a part tan.

And it acts parallel to the velocity.

So it's responsible for speeding up or slowing down the object as it moves in that circle.

Exactly.

The roller coaster example helps to visualize this.

Yeah.

At the bottom of the loop where it's moving fast, the radial acceleration pointing upward is big.

And as it goes up the loop and slows down, that radial acceleration decreases.

Right.

And that tangential acceleration is acting against its motion as it goes up.

And with its motion as it comes down.

So it's changing the speed.

It's all about those two components working together.

So far we've been talking about motion from the perspective of a stationary observer.

Right.

Someone just watching from the sidelines.

But what happens when we have multiple observers who are themselves moving relative to each other?

That's where things get really interesting.

That's relative velocity.

So the velocity we observe depends on our own frame of reference, how we're moving.

Exactly.

Like if you and I are both watching a car go by, but I'm on a moving train, we'll see that car moving at different velocities.

So how do we relate those different velocities?

The textbook starts with a simple one -dimensional case, like a passenger walking on a moving train.

Okay.

So we have the passenger, the train and the ground, three different frames of reference.

Right.

And they use this notation where Vpa at x represents the x component of the velocity of p relative to a.

So like the velocity of the passenger relative to the ground.

Exactly.

And the key relationship is this Vpa at x is Vpb at x plus Vba at x.

So velocity of p is seen by a equals the velocity of p is seen by b plus the velocity of b is seen by a.

It's like adding up the individual motions.

Yeah.

So if the passenger is walking forward at one meter per second relative to the train.

That's Vpb at x.

And the train is moving forward at 10 meters per second relative to the ground.

That's Vba at x.

Then the passenger's velocity relative to the ground.

Vp at x.

Is 11 meters per second.

One plus 10.

Makes sense.

They also mentioned that the velocity of a relative to b is just the negative of the velocity of b relative to a.

So from the passenger's perspective, the ground is moving backwards at 10 meters per second.

Right.

The textbook provides a good strategy for solving these problems.

Identify the frames of reference, label the velocities carefully, and then apply the equation.

They use that.

Two cars approaching each other.

Like to show how to find the relative velocity.

Which ends up being faster than either car's individual speed.

And then they take this to two and three dimensions where velocity is a vector.

So now we have to consider the directions too.

Exactly.

The relationship becomes Vspb plus Vsba.

But now this is vector addition.

So if our passenger was walking across the train,

their velocity relative to the ground would be at an angle.

Right.

A combination of their sideways motion on the train and the train's forward motion relative to the ground.

And the airplane flying in wind example shows this vector addition beautifully.

To fly due north in a crosswind, the pilot has to aim the plane slightly into the wind.

So the wind is pushing the plane sideways and the pilot has to compensate to stay on course.

Exactly.

And a headwind will reduce the plane's speed relative to the ground.

It's amazing how relative velocity is crucial in so many areas.

Air traffic control, navigation,

even physics research.

Okay.

So we've covered a lot of ground today.

We've really explored the ins and outs of motion.

We saw how vectors are essential for describing motion in multiple dimensions.

And how acceleration is more than just changing speed.

It's any change in velocity.

Which explains why our cyclist going around a curve is indeed accelerating.

Even at a constant speed.

We also learned that the horizontal and vertical movements of a projectile are independent.

Simplifying things quite a bit.

And we delved into the world of circular motion

with centripetal acceleration and tangential acceleration.

Always pointing towards the center and changing the speed respectively.

And we wrap things up with relative velocity.

Where the observer's motion matters.

So going back to that cyclist, they're definitely accelerating because their direction is constantly changing.

It's a perfect example of how physics can be counterintuitive.

It is.

So next time you see something moving, think about its velocity and its acceleration.

As it's speeding up, slowing down, changing direction.

And think about how the motion might appear different from another perspective.

It's a whole new way of seeing the world.

Absolutely.