High School Physics Multi-Dimensional Kinematics

Hi, welcome back to educator.com. Today we are going to be talking about multi-dimensional kinematics.0000

What happens when we are moving in more than just one dimension!0005

So, previously we dealt with everything as if we were only one-dimensional, always just one dimension.0009

Otherwise, everything is a scalar, a single number on its own.0011

But, now we are going to see some of the concepts we were dealing with before, actually are vectors in hiding.0017

All the vectors, all the ones we were talking about, are getting from one place to another place, because a place can be many dimensions.0022

We are used to living in a three-dimensional world.0030

You do not just go to the store on a direct path, not everything is always going to be a straight line, if you look at two paths or more paths.0031

So, we are going to have to talk about things using multiple dimensions, either in x-y axes or in x-y-z coordinate system, some kind of coordinate system that has more than one dimension, if we going to be talking about the real world.0039

All of our vectors are going to be the ideas that we have a change, displacement, when we are moving from one location to another location.0045

Velocity, moving at a certain speed.0057

But, more than just speed, it is going to talk about the way we are moving.0060

Acceleration, when we are changing the velocity.0064

Scalars, on the other hand were the things that were just raw length, if we changed it, it is really one-dimensional.0066

The distance between two points is, what it would be if you take a tape measure between those two points.0072

It does not care what the angle is, it just cares where, how far it is from 'a' to 'b'.0076

The displacement on the other hand, would care how did you get there, it is just not the length, there is an entire circle you could go.0082

If you were talking about some length, that length could be pointing in any direction on a circle.0090

You need more than that if you are going to really talk about displacement. Speed is similar to distance.0095

It is just distance/time, if you are looking for the average speed. That is how fast you are traveling, but once again, it is not going to say anything about where you are actually heading.0100

Finally, time.0109

We are going to treat time as a scalar, however I do want to say that if you are to get in to more heavier Physics, you would wind up seeing that time can actually be treated as another part of the space vector, where we are located.0110

That is going to have to do with the idea of space-time, that is beyond what we are talking about right now.0123

So, we are going to be able to treat time as just a single dimensional quantity all the time.0128

First, before we get started, a quick note on vectors.0134

When we are indicating a vector, I like to use a little arrow, like this guy here.0136

Other people, they prefer to use a bold font. If you are looking in a text book do not be surprised if you are seeing bold fonts everywhere, I am writing little arrows.0140

Or, if you look in one text book, and it has bold fonts, and another has little arrows, now you know why!0149

They are both ways of indicating vectors.0154

Sometimes, when we already know we are definitely talking about vectors, it is just assumed that we write it like that.0156

Finally , when we are writing it with handwriting, I tend to write it like this, because frankly, I am a little bit lazy.0161

I could write it like that, but that takes little bit extra effort, so, instead I make this little harpooned hat that lands on top of it.0166

So, little harpoon on top makes that little arrow, and that is how I write it when we are dealing with a vector, when we are writing it up.0174

But when it is written, it has that actual arrow above it. Alright! So, position.0180

Like said before, position is just the location of an object at a given moment in time.0185

But instead of having position be just along a single string, it is no longer just a single dimensional coordinate axis, we now would have to have some sort of grid, may be two dimensions, may be three dimensions.0189

We are going to have to talk about more than just one dimension though.0200

Here, we would be able to talk about the point (3,2), just the x axis distance, and the y axis distance.0203

We go over 3, and then we go up 2. Simple as that.0210

Distance and displacement. So, like before, distance is a measure of length.0216

We got two points 'a' and 'b', the distance is just that line, that straight line distance from one point to the other.0220

It is just the length that you would measure if you were using a tape measure, or walking it out with your feet.0228

The displacement on the other hand is a vector that indicates the change occurring between those two points.0232

So, 'a' to 'b' would be very different vector than 'b' to 'a', and even if we are dealing with the same length over here, but pretend it is the same length, I think it is about the same length, If we had 'c' over here, 'a' to 'c' would be a completely different vector than 'a' to 'b', even though these are the same length.0238

So, the length 'ab' = the length 'ac' = the length 'ca' = the length 'ba'.0256

But, each one of those would be a totally different vector.0262

We are talking about taking a different path, We have to go there in a different way.0265

We 6ake the same number of steps, so to speak, to get there, but we are taking a totally different path to get there.0270

Because we are going to a different final location.0276

For example, say we walk 3 km North of the house, and then 4 km East.0280

We start off at home, we go up 3 km, and then we go East another 4 km.0285

So, what would our displacement be? Our displacement will be from where we started, to where we ended.0293

So, we go like that, and that would be our displacement vector.0298

Our displacement vector would be (4,3).0301

And also notice, we are not saying this explicitly, but we almost always assume 'up' as positive, and 'right' is positive.0311

Because, that is what they are on the x-y axis on the normal Cartesian coordinate system.0321

That is what we are used to in Algebra, we tends to translate over.0325

Once in a while we might want to change which we consider to be positive, and which we consider to be negative, but for the most part we are going to treat going North as positive, going South as negative, going East, to the right as positive, going West, as negative.0329

And if we are dealing with on a flat ground, may be on a table, and had a box, if the box moved to the right, that would be positive, if it moves to the left, that would be negative.0342

If the box moved up, that would be positive, if the box moved down, that would be negative.0351

It is up to us to impose a coordinate system.0357

It is always an important thing to keep in mind.0359

It is us humans who impose a coordinate system on the world and make sense where things are, by giving them assigned values.0360

The assigned values can vary, but we have to choose how we are going to start with to make the window frame to look at the world with.0369

So, our displacement would be this vector right here, (4,3) km.0377

We traveled 4 to the East, we traveled 3 North.0382

It does not matter which direction we have done it.0385

We could have alternatively done it, it does not matter which order we do it in. We could have alternatively done like this, we would have landed at the same spot.0387

But what is the distance between where you started and where you ended? We have got a right triangle.0394

By Pythagorean theorem, we know that 3² + 4² has to be equal to, here is the symbol that we use to show distance, when we want to show how long that vector is, we use the magnitude or the absolute value as you are used to seeing in Math.0400

So, this would be the absolute value squared, so 9 + 16 = the distance squared, so we get 25, which is going to wind up becoming 5.0405

And we would technically get +/- 5, but when we take the square root, we know there is no such thing as a negative value in distance, so we know that we got to have a positive value.0429

We can just forget about the negative when we are doing this.0439

And finally, what distance did we travel? We noticed that there is a difference between the distance from beginning to end, a path a bird might take, versus the path that we actually took with our feet.0442

In this case, if we measure the feet path, we have to go 3 km North, to the first point, and then change to 4 km East.0453

So, we would be 3 + 4 =7 for the total km, for the total distance traveled.0464

Each one of these are being different things.0476

Displacement is the vector that says how do you get from where you started, to where you ended. Which path you have to take.0478

You have to, no matter how you do it, no matter what path you wind up taking, you could walk like this, then walk like this, then walk like this, then walk like this, and then walk like this, and get to the point, the same starting point.0485

That is going to wind up being up being the displacement vector of zero.0496

Because you did not ultimately displace yourself.0499

In this case, we did ultimately displace ourselves, we displaced ourselves 4 km to the East, 3 km to the North. Compare that to the distance.0502

We wound up being in a different location, we are now away from our original location by 5 km.0510

And finally, the feet steps, we had to actually walk using our feet.0517

How far we actually traveled by foot, is going to be the total distance we traveled, 3+4 = 7 km.0522

Three very different ideas, but important to remember that we can talk about each one of these things, and each one of these is going to get used, at different times.0531

Speed and Velocity.0538

Just like before, speed is how fast, it is just how fast you are going some time.0540

So, it is that length that you have traveled, divided by the time it took you to do that travel.0544

But velocity is based on displacement. It asks how you got there, not just how fast you moved to get there, but how you got there.0549

Did you go there at this angle, did you go there at this angle, did you go there at this angle.0557

Speed, since they are all the same length, if they were all the same length, that would be the exact same speed, but traveling in three very different directions.0563

We are going to have different meeting.0573

If you are traveling 60 km/h, to the North, that is very different from if you are traveling 60 km/h to the East.0575

They would be the same speed, in the pedometer in your car, but they are going to be very different velocities, because you are traveling in a different path, you are traveling in different way, different direction.0584

Velocity is based off displacement. Velocity is a vector as well.0585

Velocity is the displacement divided by the time, so v = change in displacement/time, Δ d/t.0598

Let us go back and look at the example we just had, we started in a house, we traveled 3 km to the North, and then we travel 4 km to the east.0607

We do this in 2 hours.0619

If it is 2 hours, our displacement, is equal to (4,3) km.0623

So, what is our average velocity? Average velocity is the change in our displacement.0640

Displacement is the change in the locations, we denote with d.0648

We can talk about displacement, as d, like this, but we could also talk as the change in location.0652

Once again, we have this thing where location, displacement, sometimes they get used for the same letter, so there is little bit of confusion here.0661

But we know we need to talk about how far we traveled.0667

We traveled (4,3) km as a vector.0670

If we want to find out what the velocity is, velocity = (4,3) km / 2 hours, so velocity is going to be equal to (2,3/2) km.0673

Now, if we want to know what our average speed was, we need to see what distance did we travel.0690

Distance = 5 km .0695

So the average speed is going to be, we are going to look at the magnitude, the size of that speed vector and it is going to be 5/2 = 2.5 km/h .0701

Here is an important note. Speed from velocity.0722

As we just saw, on the previous example, if you know something's velocity, we can easily figure out its speed.0724

You could figure out how far it traveled total and divide it by the amount of time to get there.0730

But you can even do better than that. Speed is the length of the velocity vector.0735

We are going to figure it out by looking at how long is the velocity vector.0741

If we want the speed, we are going to look at the distance that we traveled and divide it by the time, or we could look at the velocity vector, which is displacement/time.0746

So, if we just look at the length of the velocity, we already wound up dividing by the time, and it is going to work out.0756

Last time, we got that the average speed for that trip was, 2.5 km/h.0761

But velocity vector, v = (2,3/2) km/h .0770

So, if we want to make it a little bit faster, we just need to see what the magnitude of this is, and it is going to wind up being the exact same as this right here.0779

Let us double check that.0785

If we do sqrt(2² + (3/2)²), we get sqrt(4 + (9/4)), i.e. sqrt(25/4), which is equal to 5/2, exact same thing.0789

So, it is just a question of do we wind up figuring out the distance, and then divide it by the time, or do we wind up figuring out the velocity, which already involves dividing by the time.0818

And then just figure out how long that vector is on its own. So, two ways to do it.0826

Normally it is going to be easier if we want to find out the speed of something, we know its velocity vector, we just toss it in to this.0830

(v_x)² + (v_y)², we can easily figure out what our speed is from that.0836

Acceleration just comes from velocity. Since velocity is vector, acceleration must be a vector.0850

Acceleration = change in velocity / time .0856

So, if we have two different velocities in two different times, we find out what the difference between them is, and then we divide it by the time.0860

Gravity is going to be just as the same as before, but we remember it is only going to be effective only in one axis.0868

There is no lateral gravity on the Earth.0872

If we jump in to the air, you do not get shifted to the right or to the left by gravity.0874

So, we are not going to have any x-axis shifting and since when we normally talk about coordinate systems, this tends to be down, this tends to be up, and these are right and left.0878

Occasionally we will also look from the top-down when we talk about North and South and East and West.0890

But if we are talking about something that is lateral and moving vertically, then we are going to normally do it as up and down being the y-axis.0895

If up and down is the y-axis, then we need -9.8 m/s/s, because we are going down at 9.8 m/s/s .0901

There is not going to any real changes from the formulae before.0914

This is the exact same as it was before, except now are winding up looking at it with vectors.0919

So, everything is now working in terms of the vectors here.0924

Displacement, the location at time t = 1/2 × a t² + v_it + the initial location.0927

This one is a little bit special, in that we are going to wind up having to break it down into its components.0942

Remember, before we had, v_f² = v_i² + 2a × displacement.0948

That was what happened when we were looking in one dimension.0956

But if we are now looking in two dimensions, this equation right here applies in each dimension, so we wind up applying it over the x's and then over the y's.0959

So we just need to separate it and break it into each one.0972

We cannot talk about a vector squared, that does not mean anything.0974

We cannot just scale it, because who is going to multiply who?0977

Instead we have to break it into its components, then we easily square the components of that vector.0984

So, we just keep the x axis, the x components separate from the y components, the y axis.0991

They each do their own thing, as long as the acceleration and the distance, they are not changing at all.0996

As long as you got a steady acceleration, then we are safe, and we can talk about it this way.1001

Acceleration = change in velocity, so now we are just working in terms of vectors, and same as velocity = change in location, or the displacement.1006

Lets us look at some examples:1017

If we have a car driving North at 30 m/s, and then it turns right, so this is v_i, turns right, and the final, goes East at 30 m/s.1019

If the car takes 5 s to complete the turn, what is the average acceleration that the car has to have on it for the duration of the turn?1037

The first thing to note here, is we have to be talking in multiple dimensions.1044

We are talking in multiple dimensions, because the car is moving North and then East, so we doing in two totally different directions.1047

We cannot just do this in one coordinate axis.1054

So, what we really want to do is, we want to convert these directions and lengths into actual vectors.1056

First we are going 30 m/s to the North, so that is going to be positive, so we get, (0,30) m/s is v_i.1064

v_f = 30 m/s to the East.1078

If we want to find out what the acceleration is, acceleration = (change in velocity vectors)/(time involved).1086

That is, (final - initial)/time = ((30,0) - (0,30)) / 5 = (30,-30) / 5 = (6,-6) m/s/s , because it is changing, for every second it goes, it winds up changing either 6 m/s faster to the East, or -6 m/s 'faster' to the North.1094

What that really means is, 6 slower.1138

-6, we mean as we are now being accelerated to the South, which means since it is already moving to the North, it is going to lose some of its speed to that southern acceleration.1143

Second example is going to be a long one, it is going to create a bunch of different ideas that we are all going to hook together to help us understand how the stuff works.1155

We have a ball being thrown out of a window from a height of 10 m, with an initial speed of 20 m/s angled 30 degrees above the horizontal, we can ask some questions.1162

What is the initial velocity vector v for the ball?1170

Then, ignoring air resistance, how long does it take the ball to hit the ground?1173

Finally, ignoring air resistance once again, what is the ball's displacement from its starting point, and what is its distance?1177

We have got some building, and a window, somebody throws a ball out of that window.1185

We know, if we were to set a horizontal straight line, this is going to be an angle of 30 degrees, and this ball comes out at 20 m/s.1194

Down here is the ground, and as time goes on, the ball is going to fly forward, and then gravity is going to take more and more of a share of its velocity, and it is going to eventually land, hit the ground somewhere.1204

We want to figure out how long is it going to be flying through the air, how long is it going to take before its y location hits zero.1218

Once we know that, we can figure out, how far is it going to make it to the right.1228

First, we are going to have to know, what is its initial velocity vector.1232

If we have got, something that is 20 long on this side, and 30 degrees here, we can figure out what the other sides have to be, just use trig.1234

So, this side, since it is the side opposite, is going to be sin(30) × 20.1247

This one, would be cos(30) × 20.1254

So, cos(30) × 20 = sqrt(3)/2 × 10 = 17.3 m/s, approximately.1259

And sine is going to be 10 m/s.1272

That is going to give us initial velocity vector of (17.3,10) m/s. (right, up).1276

Now we are ready to move on, because now we have got a vector, and we can make use of this vector, and break each component off, and work with each component separately.1289

If we want to figure out how long does it take the ball to hit the ground, we do not care what its x component is, it is the same thing if the ball hits the ground here, or here, or here.1296

All that matters is, when does it have that zero.1306

When does the ball make contact at the ground, when is the y axis location zero.1310

If we have got, d(t) = (1/2)at² + v_it + initial location, we can then just break this into looking at the y's throughout.1316

We just look at the y components throughout, and we can figure out when it is going to hit the ground.1333

What is its acceleration?1338

The acceleration is, -9.8, so (1/2) × (-9.8) × t² + 10t + 10 .1343

We want to solve this for when does its location becomes zero.1363

We want to make that left side set to zero, and we want to see what 't' will allow that zero to appear on the left side.1366

So, -4.9t² +10t +10.1375

How do we solve something like this, we got a couple of choices.1379

Right here, we got a polynomial.1383

If we have got a polynomial, one of the first things we can do is factor it.1386

To me, that does not look really easy to factor.1390

The easiest thing after that, is to chuck it into the quadratic formula, make that machine go through it.1392

What is the quadratic formula?1400

(-b +/- sqrt(b² - 4ac)) / 2a , and that is going to be the solutions when time is going to make it equal to zero.1402

That gives us one way to do it, one other way, you could just plug that into a computer or a powerful calculator.1414

But let us go through the quadratic formula, because we could all just work through it that way.1423

The answers, t = (-10 +/- sqrt(10² - 4 × (-4.9) × 10)) / (2× (-4.9))1427

Keep working through that, we get, (-10 +/- sqrt(100 + 196)) / (-9.8) = (-10 +/- sqrt(296))/(-9.8) = (-10 +/- 17.2) / (-9.8)1455

Now, we are going to have to ask ourselves, are we going to go with the plus, or are we going to go with the minus.1490

Because mathematically, both of those are correct answers, both of them are times when this equation is going to be fulfilled, it will be fulfilled, that equation will be fulfilled both at the plus and the minus.1493

But, we can see logically that the ball only hits the ground one side, it does not have both a forward time and a negative time.1511

So, what we want to do, is we know our answer can only be a positive time, because this equation is not true at t < 0.1520

Before this moment, we do not know where the ball was, the ball was sitting in some apartment before somebody picked it up, and decided to through it out the window.1528

At that moment, that equation right here, was not true before time = 0.1534

Once time = 0, that is the moment the ball is actually thrown, we know we can start using this equation.1540

So, the only solution that will work, is the one where time > 0 .1546

We look at this, and we are going to have to find a way for us to have a negative number on top, because (-10 + 17.2) will give us a positive on top, and then we divide by a negative, we wind up getting a negative thing.1549

We are going to actually have to go with the negative answer right here, not the negative answer, but the negative of the plus-minus, because it is the only one which would work.1562

If we were to solve it out and get both answers, we would be able to get 2 of them, one positive answer and a negative answer.1571

But then the negative answer we know, cannot work, so we have to chose which sign is going to give us a time that is positive.1579

We chose the negative one, so knock out the positive one, so we are going to get, 2.78 s.1587

So that ball has 2.78 s flight time, before it hits the ground.1597

With that, we can now figure out what its x location is, when it hits the ground.1602

If it has got 2.78 s flight time, we have see how far it has managed to travel in the x, before it hits the ground.1606

Its location in the x is just going to be its 17.3 × 2.78 .(velocity × time)1614

But, do we have to worry about acceleration?1639

The formula that we normally work with, (1/2)at², remember, there is no lateral gravity, we do not have anything accelerating the ball once it is thrown, so all we have to worry about is the initial velocity.1642

We multiply those out, we get, 48.1 m is the x location, so if we want to combine that, we have got initial location vector is at where, where does it start?1653

Does it start at (0,0)?1673

No!, remember, the house starts up here, 10 above the ground.1674

So we will make the x axis have it be zero at the point of the window it comes out of.1678

So zero for the initial location, but it starts at +10.1683

Final location, it hits the ground, and it is at 48.1, and then it has got 0 here.1688

So the change, the displacement that it experiences between those two locations, is going to be (48.1,0) - (0,10).1697

So we are going to get (48.1,-10), because it traveled down 10 m before it hits the ground, is the displacement it experiences.1711

If we want to know what the distance it experiences is, remember, we throw it out of the window, and it lands over here, if we want to know what its distance is, we just have to measure, there is the displacement, so we just need to know what is this length.1722

So that length is going to be, the size of the displacement vector.1737

Size of the displacement vector, sqrt((48.1)² + (-10)²).1744

Punch that out, we get, 49.1 m1754

So the ball has a displacement vector of 48.1 to the right, and down 10, so -10 on the y axis.1761

But the distance between where it lands, and where it started moving is 49.1 m.1769

If we want to know how far its travel was, we do not have the mathematical technology yet, we will be able to figure that out.1773

We need to go learn some more calculus before we will be able to figure out what its travel path was, how long it had to move through to get where it landed.1781

But we can figure what is the distance from where it landed to where it started.1790

Hope that all made sense, see you later.1795