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0:00
[MUSIC PLAYING]
0:08
This is a PVC pipe about 2.5 meters in length. Inside of it is a ping-pong ball on one end. We turned on a vacuum pump, which proceeds to suck out most of the air. When that happens, the pipe is now a potential cannon. But how? Because we can puncture the thin polyester covering, and all the pressure of the entire atmosphere will push the air in and accelerate the ping-pong ball. From just this footage, can I figure out how fast the ping pong ball shot out the end?
0:36
Hello, I'm Dianna Cowern, creator of the YouTube channel Physics Girl, and welcome to lesson one of Dianna's Intro Physics Class, also known as AP Physics One Review, also known as Physics by Dianna. Today's lesson? Kinematics. I asked a friend to define kinematics, and he guessed that it was the motion of the human body. He was close. Kinematics is the physics of motion in general-- how things move; not just humans. So that's what we are doing today.
1:03
So you can probably guess, kinematics is really important if you're trying to move through the world without getting hurt. So today's theme is to be safe or not to be safe. But show don't tell, they say, so let's show what kinematics by jumping right into a question, which is, can you design a safe-- can you design a safer bike helmet? I could have used that earlier this week. To work up to the tools we need, we need mathematical modeling. And so we begin our mathematical modeling section.
1:41
[LAUGHTER] Here's a simple scenario, a person mountain biking on a flat trail. [LAUGHTER] That does not look like a bike. Let's give her some hair. Oh, geez. [BIKE BELL RINGS] She's going 10 meters a second. She's a very steady, safe cyclist, which I guess is a nice thing to say about someone, someone boring. Side note here about units. You're probably used to going miles or kilometers per hour, not meters per second. But the math is a lot easier in metric units,
2:08
so there's a really simple conversion, where you just double your meters per second, and you get miles per hour. So 10 meters per second is about 20 miles per hour. You can check that on Google. It's not exact, but it's pretty close. OK, our cyclist, she could start anywhere, but we're going to start her here at position equals zero. And we've reduced her to a point particle. Yeah, physicists do that a lot. A second later, she's gone another 10 meters. This is my time, and this is my position.
2:34
A second later, another 10 meters. A second later, another 10 meters. The graph would look like this, where my horizontal axis is time, and my vertical axis is her position in meters. There she is. We've made a simple mathematical model of her motion. Pause everything. Here is a key wonderful moment, because this is our first solid example of what we mean by mathematical model. That's what we talked about in the intro video to this whole class, the fact that physics makes mathematical models.
3:02
And if you're going, um, what? There's no math here, I'm going to say, what you talking about? Because math isn't just xs and ys, three plus two equals six, which isn't even true. Just checking if you all are paying attention. Graphing is math. Geometry is math. OK, a graph is really powerful, because you can use it to predict the future. No joke. Predicting the future is way common in physics, like this. I drew a line through these points. I can see where my mountain biker is
3:26
going to be after six seconds. This line predicts the future for as long as she keeps going that direction at the same speed. But let's talk a little bit about what a model is and what it is not. It is a simplified representation of what actually happened. Remember the point particle? That was a simplified representation. Take, for example, breaking a wineglass with your voice, which is possible. I did it in an old video a couple of years ago. I had to sing the resonant frequency of the glass
3:52
over and over. Really annoying. E sound. [HIGH PITCHED SOUND] And then the glass broke. [GLASS CRACKS] [CHEERS AND APPLAUSE] And then, I can graph this simplified model. Something like this, where if I match that frequency right there, I get a really high amplitude response, and my wineglass shakes and shakes and shakes until it breaks.
4:16
There's a lot more physics going on than this graph shows, but I can use it to pull out the most important information. So this is simplified. It is highly unlikely that the cyclist biked through every single spot on this line. She might be a boring cyclist, but she's also an imperfect human. Kind of mean to her. She'll speed up and slow down, so her trajectory will look more like that. But if we're trying to figure out, for example, how soon she's going to get to the edge of a cliff-- "Ahhh!"
4:43
bicycle wheels, girl, we don't need to know her position at every single nanosecond. A mathematical model lets us ignore extraneous details and tells us on average what she's doing. So therefore, a mathematical model is simplified. Say it with me-- simplified. But even so, I can do a lot with this graph. For example, what if she sped up? This would be her motion. A steeper line. What if she slowed down? This would be her motion. Less steep line.
5:14
What if she started here at x equals 30 and just sat on the trail eating trail mix? Well, this straight line would be her motion. What if she was biking backwards? This would be her motion, into the negatives. That's pretty ineffective for winning a race. Her choice. This graph looks simple, but it contains so much information if we look at specific things, like the slope. Let's just draw this again.
5:43
La, la, la. OK, mathematically, slope is rise, like this, over run, like this, which is often written as delta y over delta x. But here, our rise is delta x position and our run is delta t, the time. So the slope of this line is our change in position over our change in time, which equals velocity. I look at this graph, and I can tell how fast she's going by looking at the angle of this line. Let's check that with units. Here's another aside. Throughout this course, we're going
6:14
to pay close attention to units. My high school teacher taught me this trick, and it's called dimensional analysis, and it helped me solve so many problems during my college exams at MIT--I kid you not. We are not going to write down an equation in this course without checking the units to see if they make sense. Physicists use dimensional analysis all the time to make sure that we're not adding apples to oranges or pie to cake. Sounds really good, though. So let's think about our units here.
6:38
Delta x, which will be meters, and time, which will be seconds, equals-- and the units of velocity are meters per second, because it's a rate of distance.
6:49
Checks out. Did you guys know about this? Dimensional analysis? Did you do that in like your high school class? It's so useful. I swear. Now, the graph we drew is a model of her motion, but the algebraic form of the graph is also a model of her motion, and it's a powerful one, because you can manipulate algebra. Check it out--looking at the slope of this graph, I've defined it as delta x over delta t and called that velocity. If I multiply both sides by delta t, these cancel. Oops, just kidding.
7:20
That doesn't cancel. I have a new equation. Delta x equals v times delta t. Just by doing that simple trick, we can calculate her change in position. We have a new tool. This is a new tool. Now, I can predict where my cyclist is going to be at any point in the future as long as she doesn't change her velocity. In the original graph, we have her changing her position by 10 meters every second. So her velocity is 10 meters per second. I want to know where she is seven seconds later.
7:49
Let's check the units. I've got seconds divided by seconds. Those cancel out, and I'm left with meters. And she will be 70 meters down the road. There it is. We figured out how to model someone moving with constant velocity. Mission accomplished. Plus, we found that in a position-versus-time graph, the slope of the graph is velocity. And we pulled out our first simple, useful equation. Great. Now, moving on, there's another useful way to model this same motion, which will allow us to solve
8:17
slightly harder problems. Instead of graphing her position versus time, what if I graphed her velocity versus time? Oh, geez. So that's velocity. This is time in seconds. She's moving at 10 meters a second, and you'll notice that is a horizontal, flat line. She's probably more like this. But we're going to model it as a constant velocity. If you zoned out for a second, you might notice that our y-axis has changed. It is v now for velocity. It's not x anymore for position.
8:51
The horizontal axis is still t for time. This is one second, two seconds, three seconds. And now, I want to know where she is at four seconds. And I make a rectangle here. Big old rectangle. I want to get the area of this rectangle for a reason. You'll see why in a second. Area equals length times width. In this case, the width my rectangle is my time, four seconds. And the height is my velocity, 10 meters per second.
9:22
The rectangle's area is exactly the same equation that we just derived for her change in position, velocity multiplied by time. Now some of you are screaming, "Integration! Differentiation!" And you are right. Those of you who have no idea what I'm talking about, it doesn't matter. These are tools from the math called calculus, and the tools are called integration and differentiation, which is all about areas under the curve. That's why I'm bringing it up, but you
9:45
don't need to understand calculus to do physics, which is cool. So anyways, here's the takeaway. The area under the curve is your delta x. Delta x is your change in position. It's where you are minus where you started. So in this whole problem, we just graphed the same exact motion as the previous problem, but this time, we switched from position graphs to velocity graphs. Why did we do that? Why did we change strategy? Well, you're going to see. We just showed that the rectangle,
10:09
that area under the curve, is displacement. So now, we can use that displacement for more complicated motions. So let's go on. Let's do it. What if her velocity isn't constant? What if she's accelerating? Say she's driving on the highway starting at 20 meters a second, but then she starts to accelerate. And a few seconds later, she's going 30 meters per second, about 60 miles an hour. Let's assume that she accelerates pretty smoothly, basically a constant acceleration,
10:31
and it took her four seconds to do that. That's her motion. To find her distance during those four seconds, I'll repeat what I did for the last graph and calculate the area under this curve. Look, a rectangle and a triangle. I'm going to use elementary geometry, my dear Watson. Let's start with the rectangle. I can write the change in position for that part of her motion is just her starting velocity, because that's the height, which I'm going to write as a v with a little zero,
10:54
and I'm going to call that v sub not. I'm putting a zero next to that velocity because she's changing velocity, so this is her initial velocity. And the area under this rectangle is v not times t. In this case, v sub not is 20 and t is 4, so the area of the rectangle is length times width. 20 times 4 is 80. My seconds cancel. 80 meters. Let's think about what that means for a second. If she weren't accelerating, the rectangle is the distance she would have traveled, just like before.
11:21
She would have gone 80 meters. The triangle up here, though, this is new. It's the addition to her travel distance because she's changing her velocity. She is accelerating. Let's look at the area of this triangle, which is equal to her change in position while she's accelerating. Well, the area of a triangle is just one half base times height, and we're going to see something really cool. The base is time, and the height is the difference between 20 and 30 meters per second.
11:46
That's her change in velocity, or delta v. Doing the math in our heads real quick, the change between 20 and 30 is 10, times 4, times a half, is 20 meters. So how far did our cyclist go in those four seconds? Well, she went 80 plus 20 is 100 meters.
12:08
The rectangle, that's our v not times t. The change of position because of her acceleration, that's 1/2 t times delta v, and so how do I change my velocity? By accelerating. Acceleration is change in velocity over change in time. Let's make sure the units work out. The units of velocity are meters per second divided by time, which is seconds. I get a new unit, which is meters per second per second, or meters per second squared, and we get a new unit for acceleration.
12:32
It's not a very intuitive unit. So for some context, things would fall on Earth with an acceleration of 9.81 meters per second squared if there was no air. The acceleration you'd feel in a car going from zero to 60 in three seconds, 8.9 meters per second squared, which is not even as much as gravity. We can look at the acceleration of the mantis shrimp's claw, which is an insane 3,000 meters per second squared, which is ridiculous, as compared to fast humans running,
12:58
who accelerate at about three to five meters per second squared. And the theoretical fastest my ping-pong ball can accelerate in the tube is about 49,000 meters per second squared. We'll see how fast it actually accelerated, though, when we get to that problem. We can use these shapes to come up with an equation for distance traveled. Here's something fun. I'm going to do my third favorite trick. I'm going to substitute something in. Notice these two equations that I wrote down--
13:21
the area of this triangle, 1/2 times delta v, and then, this over here is just a simple definition of acceleration. Notice, both of these equations have a delta v. I'm going to rewrite this one doing exactly what I did before bringing up the delta t. These cancel, and I see that delta v is just acceleration times time. It's a simple little algebraic substitution. When I see two different equations have something in common, maybe I can combine them. And if I plug this in here, I get
13:46
something really interesting. I get that the area of the triangle, this here, due to my acceleration is now 1/2. Bring it up--1/2 times delta v, at, and then another t, at squared. Now, I'm going to put all this together. I'm going to put together the area of my rectangle and the area of my triangle. That's 1/2 at squared plus-- where's the other one? V not t.
14:12
And in this problem, what were we doing? We were doing the change in position, which has two pieces. It has how far I would have gone if I wasn't accelerating and how far I went because I accelerated. And remember, I don't need both. Maybe I have a case where I'm not accelerating, and this term goes away. Maybe I have a case where I had zero initial velocity, and this term goes away. I could have a case where I'm doing both. I could have a case where I have an initial velocity,
14:32
and I'm accelerating. And then, I need both. Now let's rewrite this, but we're missing one key term, the initial position. My initial position here was zero, so I didn't need that. But if I add it to these two terms, I get position equals initial position plus initial velocity times time plus 1/2 times acceleration times time squared. You may have already seen this equation a bunch. We call this the equation of motion. Ahhh! Where you are is where you started
15:01
plus how far you went due to your initial velocity plus how far you went due to the fact that you're accelerating. I bet you never understood this equation so well. We have, now, one simple, master equation-- the equation of motion. This equation is important. Write it down. Tattoo it on your arm. And we just derived it. We just looked at the area under a velocity versus time graph to find displacement, and from that, we derived the equation of motion. That's pretty cool.
15:26
You can use this to derive all the other equations you have in your book or cheat sheet. Here they are, if you want to memorize them. [MUSIC PLAYING]
15:37
And I can now solve my ping-pong-ball question. How fast is the ping-pong ball going when it leaves the pipe? I only need two measurements, the distance the ball traveled, which I measured using the nice tape measure on screen and I got 2.29 meters, and the time, which I measured by knowing this was filmed at 18,000 frames per second. So each frame is 1/18,000 of a second. And I get about 0.0121 seconds. So my average velocity is 2.29 meters over 0.0121 seconds
16:05
distance over time, which equals 189 meters per second. If we assume constant acceleration, my final velocity is twice that, then, at 379 meters per second, which is over 800 miles an hour. Dude, that's faster than the speed of sound, which doesn't seem right. So I redid the same type of measurements, 4.5 centimeters of pipe, which only took three frames, and I got 270 meters per second, which is a lot more reasonable. So why the difference? Well, it's not actually constant acceleration.
16:36
There's a bunch of complicated things going on, including the fact that air is rushing in, not just pushing the ball. It's also pushing itself. But I did measure an incredibly high acceleration of 47,000 meters per second squared for the first 10 centimeters. Can you think of how I did that? And voila. Now, my final question is, can you build a safer bike helmet? OK, say you've got a mountain biker. She's wearing a helmet, and she runs into a tree going around 10 meters per second,
17:02
about 20 miles per hour. And she loses her balance and falls off the bike. I'm a much more careful cyclist. I want to know what her face feels. Specifically, I want to know what the acceleration of her head experienced. Her head comes to a complete stop in the distance that part of her helmet crumbles, which we'll generously say is about 10 centimeters. Now, I don't have any fancy laser devices, I just have her speed and the helmet crumple length, 10 centimeters.
17:29
If I just plug this into my equation of motion, which we'll write again, this term goes away, because her initial position is just zero. I know her initial velocity. I don't know her time, and I don't know her acceleration. I have two unknowns. Now, there's probably some fancy equation in your book you could just plug and chug, but to me, that gives no intuition for what's actually going on. So let's do this. I have this equation for x, but I have unknowns, a and t.
17:51
So how can I find one of them? Well, I know that I started at 10 meters a second, and I stopped at zero. So my change in velocity is 10 meters per second. So let's assume a constant deceleration. We'll just graph it. That means you go from 10 to zero. This means I know my average velocity, because it's just halfway between 10 and zero. It's five meters a second. So my average velocity is five meters per second. And I know I was at the average velocity for 10 centimeters, or 0.1 meters.
18:21
So then, I knew how long it took me to stop. I know the time, because velocity is change in position over change in time. All I have to do is rearrange to get the time. This comes up here. This goes down here. X over v. I know this is 0.1 meters, and this is 5 meters per second. And I see that the time it took me to stop is-- let's go over here-- 0.05 of a second. Now, her acceleration. Well, it's the change in velocity, which is 10 meters per second, over change in time, which is 0.05 seconds.
18:54
And I get 500 meters per second squared. That's about 50 times the acceleration you would feel from gravity here on Earth's surface. So that's a lot. But it's survivable. You may have heard of this sort of comparison for acceleration called gs. So for example, my friend Destin from the channel Smarter Every Day got to go up in an F16, and he experienced 7 gs. Hoping it happens one day. The highest ever recorded gs experienced by a human were 216 gs. Want to hear something crazy?
19:27
The mantis shrimp punch-- that's the majestic creature that I made a whole Physics Girl video about--its claw accelerates up to around 15,000 gs. So now, you're thinking to yourself, thank goodness she was in a bike accident and not punched by a mantis shrimp. So we found the bike accident is survivable, but you might still get a concussion. So how do you make your helmet safer? Well, if you increase the distance you need to stop, say if you have an inflatable bike helmet that gave you
19:55
another 10 centimeters, well, that's like how the crumple zone, the front of a car that crumples when you're in a collision, helps you feel less acceleration. So you can work out the inflatable helmet problem. Go Google inflatable bike helmets. Figure out what the crumple zone of a helmet would be and work out the acceleration. Send me the answer if you do it. I believe in you. So that's our first lesson on kinematics. When they ask you what you learned on YouTube today,
20:19
here are your two important takeaways-- If you can turn your physics problem into a graph or a picture, do it. If you can find your average velocity in problems with 1D constant acceleration, use it. And here are all the problems we did today. You know why? Because the only way to really learn physics is to work problems. So here's your homework. Go work all these problems again on your own. For real. I'm not kidding. Go do them. You can also find a ton more one dimensional motion kinematics
20:45
problems online. So if you find any other interesting problems, post them in the comments. We've just touched on the basics of kinematics. Because if you stick with physics, stranger things happen. For example, in this whole lesson, we assume that time passes at the same rate for everybody, that one second for the cyclist is equal to one second for the tree she hit. It's not fake physics we just did, but it's a less precise approximation. If you want to be really precise,
21:09
time actually passes more slowly if an object is moving. And you really notice it when that object is traveling close to the speed of light. That's real physics. You get cool thought experiments, like where one twin leaves earth for two weeks and comes back to find that their twin has aged 40 years. If you continue on to take physics and take a class on special relativity, you'll learn about that craziness and a whole lot more. And I hope you do stick with physics.
21:33
And now a message for you from a special guest. Hi, my name is Simone. I run a YouTube channel named after myself, because I couldn't think of anything more fun. But little known fact is that I actually used to study physics in college. I did drop out after just a year, which is a little bit of a parenthesis, but I love physics, and I am so excited for you to learn more about it, because what better way to understand the world than through the magic of numbers? Good luck. [MUSIC PLAYING]
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