[MUSIC] Welcome back. This week we'll be describing motion, which is called kinematics. For this we need to quantify time, position, velocity and acceleration. Later on, we'll explain the motion by analyzing forces, which are the causes of accelerations. This week, we'll analyze motion in one dimension. We'll also look at relative motion and frames of reference. For both topics, we need to remember that vectors have magnitude and direction. While scalars just have magnitudes. For example, in week one, we distinguished displacement, a vector, from distance, a scalar. And also velocity, a vector, from speed, a scalar. Here's a question to remind us of the difference. At the Sydney Olympics, Cathy Freeman not only lit the Olympic flame, but also went on to win the 400 meters event in the rain, in a time of 49.11 seconds. Think about her average velocity for the event, then answer this question. What was the magnitude of her average velocity? Aha! We've learned something. What about this one? I'm cycling at fairly constant speed, but my velocity, velocity's a vector so it has magnitude and direction, my velocity keeps changing. I'm travelling south, west, north, then east. We'll leave circular motion til next week. But remember that because my velocity is changing, I am accelerating even though my speed is constant. Any change in velocity is an acceleration. For now, we'll look at motion in a straight line. 400 years ago, the Italian physicist Galileo Galilei founded the science of mechanics with very simple equipment, such as you might use in home experiments. [SOUND]. Here's an example. Measure X along the board with 0 here and a T from when the ball is released. We can see it's getting faster, both displacement and velocity are increasing. We can plot the X displacement from 0, the X coordinate as a function of time, T. Here we're plotting the position of the center of the ball. With macroscopic objects, we need to choose a representative position, what point on this object would we choose? For the purposes of this course, we shall neglect internal motion and we'll give an object's position as just a point in space. Technically, this means we are treating them as particles, which is part of the name of the course. Now back to our rolling ball. Here's displacement versus time. Over this time interval, the average velocity is just the displacement divided by the time taken. Here x over t, which is 1.14 meters over 4.2 seconds, which is 0.27 meters per second in the positive x direction. Here we can also estimate the time at which the ball had a velocity equal to the average velocity. Looks to be about 2.1 seconds. We can measure average velocities over smaller time intervals, too. We use the Greek letter delta to mean the changing. Write this a few times so you get used to it. Delta x is the change in x. So the average velocity over this small time interval delta t, is delta x over delta t. Here's a plot of the average velocity measured over single frames. Well, there are substantial arrow bars because of experimental uncertainties. It doesn't move very far in one frame, so the proportional error in velocity is large. [LAUGH] Yes, you've probably already found that out if you've started the first experimental exercise. It's possible to measure things much more precisely, of course, with different technologies. But uncertainties in measurement are always present, so we can never know precisely what the so called instantaneous velocity is at any given time. We can however fit smooth curves to the data. But, remember that those curves are only theoretical approximations. Here's a simpler case. The air track is level and the speed and velocity are both very nearly constant. With this gadget, the cushion of air prevents contact and removes friction. Using the scale for distance and camera frames to measure time, we plot x versus t. Question for you. What is the magnitude of the average velocity between t equals 0 and t equals 1.6 seconds. Good. Using the first and last data, the average equals x minus x0 over t. Which is one of the important equations in kinematics. Write it down and remember it. In this experiment, it looks as though the velocity is almost constant. Let's plot the velocity as a function of time. Subject to that constant velocity approximation, we can write v average equals v, equals x minus x0 over T. So that shows us how to get from displacement to velocity we take the slope. Velocity is the slope of the displacement graph. Going in the other direction we can rewrite v equals x minus x0 over t, and rearrange it to give x equals x0 plus vt. This shows me how to get from velocity to change and displacement. The term v times t, is the area under the velocity versus time graph. I've used positive v to mean traveling in the positive x direction. In contrast, here's our negative v, traveling with the same speed but in the negative x direction. A question for you. Peter leaves home and walks east at a constant 5.0 kilometers per hour. At the moment he leaves, his friend Kate, who lives 6.0 kilometers directly east, leaves her home. She heads west, bicycling at 9.0 kilometers per hour, also constant. Note that I add the given information to my sketch. Note also that Peter's velocity is positive, but that Kate's is negative. Her speed is 9.0 killometers per hour, but she's traveling in the negative x direction. So, negative x velocity. Well, you've guessed the question. When and where do they meet? Let's do this formally, noting all the steps. Yeah, maybe you can do this one quickly in your head. But it's good to get practice in showing all the steps so that you can still do this logically when all the problems get trickier. Great. You've just solved simultaneous equations in kinematics. But, one more possible complication. What if the two objects don't leave at the same time? Here's one. Jack leaves home heading in the x direction and walking at 5.0 km per hour. Maria leaves the same point but starts 0.50 hours later. She's bicycling in the same direction at 9.0 kilometers per hour. When and where do they meet? Let's try doing this two ways. First by algebra and then with a graph. Okay. Let's see that on the displacement time graph. Here is x for Jack, leaving at t equals 0. And here's the x for Maria, leaving half an hour later. And you've just solved for the intersection of the two lines. Time for a little quiz about velocities. And also, if you need it, an introduction showing how to use spreadsheets to do your arithmetic for you.
