https://youtu.be/dpxUbdA7kDk
PHYS 1101: Lecture Four, Part Four
Now let’s move on to the second quantity we need to think about, and that’s how fast and in what direction, and we’re going to use a velocity vector to do this. The magnitude of this velocity vector is going to tell us how fast and the direction, obviously the direction that that vector points. The velocity vector is always going to point in the direction that that object is headed at that instant.
First point here to emphasize, if you think about trying to appreciate how fast something is going, you necessarily have to consider two instances of time to learn anything about that. If I just take a picture, one snapshot of an object, I have no idea if that car is parked, if it’s going forward, if it’s going backward, or how fast going. I really need a movie. I need at least two frames, and I need to know the time interval between the two, to actually see how far the object moved during that time interval, to tell you anything about the velocity.
So, the velocity is specifically a vector quantity that tells us how the position is changing with time. Let’s consider here a scenario below where let’s say I have an axis defined, positive direction, and an origin, and I observe over some time interval that at time t0 the object was here and at a later time, the object is — let me make it here. Then these two spots that I’ve drawn would be at two spots on a motion diagram, so I am comparing a position vector at one instant to a position vector at a later instant, and a vector that I would draw or could draw between these two points could be a vector that could represent this average velocity.
It has some length that’s going to tell me about the magnitude of the speed. We know if the dots are further apart it’d have to be going faster, and obviously the direction tells me between these two time intervals the object did move from location 1 to location 2.
So, the first quantity I need to define with respect to velocity is the average velocity. The average velocity is the displacement divided by the time interval, so in my picture up here, if this was time t0 and this is time t, the displacement vector between those two I know is a vector that points in the same direction as this velocity and goes from this position to this position.
Displacement here, if these are tick marks representing single meters, so this is 0, 1, 2, 3, 4, etc., meters, then I know that my displacement Δx is +3 meters, and let’s say that our tick marks here in our motion diagram are separated by 1 second. So, t minus t0 is 1 second. Let’s just use those numbers as an example.
Let me also write more specifically for you that this velocity, v, it’s equal to displacement over time, and let’s go ahead and substitute in what that is. The displacement is the final position vector minus the initial. It’s the change in position divided by the change in time.
Okay, first bullet. We’ll work with these numbers as an example, but I want to slowly go through and be sure it’s very clear with this vector where and how the magnitude or value of this is defined and the direction.
Okay, here I’ve written it again. The average velocity is equal to the displacement over t minus t0. Another way that we write that is Δt. Let me circle that up here for you. A final minus an initial, Δt, the change in time.
Okay, for this velocity vector, let me draw two little lines here because at the top we’re going to specify the magnitude. Where does that come from? The magnitude of this vector, the speed, it’s a number, and it comes from the magnitude of the displacement divided by the time interval. This isn’t a vector. There’s only vector information in the top here.
So, we call the magnitude of the velocity vector the speed, and that magnitude is set by the magnitude of the displacement vector divided by the number of seconds, that time interval Δt. It’s going to have units of — what would be the units of a displacement? Length, and so meters, divided by seconds. Meters per second. We convert that to SI units.
Okay, that’s the value. Where does it get the direction? The direction of the velocity vector, as I’ve said, is always points in the direction that the object is headed. So, let me write direction is where object is headed. Okay, for every one of these vector quantities that we define, I want to clearly draw this out for you because you need to have a physical picture of where that vector gets its magnitude and its direction from the physical problem.
I have a comment here on the book syntax to help hopefully avoid some confusion. The book uses bold print to indicate a vector, and it draws just a straight line over the top to mean average. When I’m working with average velocity, I’m just going to use a v with a single arrow over the top but just make it clear by saying “average velocity” if that’s what I’m working with.
Okay, the average velocity, while it’s important and useful to us, a lot of these problems we’re going to find that the velocity is changing, an object is speeding up or slowing down, so we’re going to need a more specific way of indicating that or at least noting that at each instant of time I have a specific velocity.
One way to think about this is as you step on the gas and you’re accelerating in your car, your speedometer, it’s telling you your speed, and the direction that your car is headed would be the direction. The combination of speed and that direction you can imagine then is the velocity vector for you in this car.
As you glance down at the speedometer, at every instant you’re going to see the needle be at a certain value, and that’s going to be sweeping along or changing as you accelerate. So, at each instant your velocity vector is different. The speed is changing.
So, we’re going to have the need to think about this velocity at an instant. We call that this instantaneous velocity, and that’s what v, the symbol v with an arrow over it, is going to represent in those instances. It’s the velocity at a specific time.
Mathematically, we can think about this instantaneous velocity as being the same mathematical equation that we wrote down for average velocity but just for very short time intervals. The average velocity is the displacement over time, and if we think about that time interval as getting very very small, then the displacement of course becomes smaller and smaller during that small time interval, but that ratio mathematically does converge to the velocity at that instant.
For those of you that have had calculus, you have faced this notion and this thought in your calculus class. Velocity in an instant really is the displacement divided by a time interval, but it’s in the limit that that time interval gets very small.
Okay, here’s a little movie, a short movie that I want to show you to emphasize this difference between the average velocity and an instantaneous velocity. So, here’s a movie of the same little block on air track, and it starts out here with the track being level, and so the motion is a constant velocity initially here.
Now what they’re going to do is tilt up one end of this track, so this block is now going to accelerate down to the right. So, in this motion diagram, you can see that the spacing between these dots gets larger and larger, conveying that the velocity’s increasing, the block is accelerating.
I know the movie’s a little bit choppy for you, but you could probably picture in real life that the motion would be very smooth as this block very continuously accelerates down the slope right from the instant it’s let go or it bounces off until it hits the other bumper at the other end.
So, if I went in and measured between any time interval here, let’s take these two, this distance, that would be the displacement between this instant and that, and then I divided that distance, that displacement magnitude, by the time interval, that would give me the average velocity here, but in real life you can picture that this object moved continuously and that, in fact, during that brief time interval I had a continuous range of velocities that started at this instant a little bit slower, and then the object was going a little bit faster by the time I hit this interval. It didn’t have a constant velocity between these two. That’s a distinction between velocity at an instant and the average velocity. Hope that helps.
Okay, here I’ve just drawn for you on top of a snapshot of that movie of one of the frames the information I was trying to emphasize here for you, that you could talk about the average velocity between any time interval. The magnitude of that would be the meter displacement between those two instances divided by the time interval. But we know in real life that between this snapshot and this snapshot that there was a continuous motion of the object in between, and at every one of those instances I can talk about the instantaneous unique velocity.
This next quiz question is asking if you have a physical feeling for what the meaning of velocity is. So, they give you this question where a train is moving along at a steady 30 meters per second, so every second the train is advancing by 30 meters. At t equals 0 it passes a signal light and that’s where our origin is defined, x equals 0.
So, without using any formulas or the equations, what would be the position of that train at t equals 2 seconds? If you need to, draw a little sketch here off to the side so it’s clear to you. Where’s your origin? x equals 0. That’s at time t at 0 seconds. It’s moving, make it moving to the right. What’s the position coordinate at t equals 2 seconds given where your origin is defined?
Let’s go through the same type of thing we did above. First, let’s be clear on how we’re going to mathematically write or express this velocity vector. We’re going to write that the velocity is equal to the scalar component v times the unit vector. If it’s right-left motion, this unit vector will be our x-hat, our x unit vector, if its up-down, this would be y. The key, though, is we are going to work with the scalar component. Let me write it out here for you.
And you have to be thinking about there will be the sign that’s associated with that is going to indicate the direction of this vector. Remember, the direction of the vector, if it’s velocity, represents the direction the object is headed.
In my motion diagram, if my velocity vectors point to the right, the scalar component has to be positive for that instant. Likewise, negative would be the object heading to the left. Its velocity is to the left.
So, let me make that clear here. Positive x-axis, if we define it to the right, in my motion if this object was moving to the right and a velocity vector would be drawn to the right, in which case v, that scalar component, I would have to make it positive. Similarly, if I had an object and it was moving to the left, that’s in the negative x direction. My scalar component for v then is going to be negative.
If I use the sign on these scalar components to indicate direction, we’ll see in a moment that the equations that I can work with to predict where an object’s going to be or how fast it’s going to be going will be — I’ll be consistent with the meaning of the sign, and they will allow us to describe objects moving to the right or to the left, even objects that slow down and turn around, etc.
So, here’s my variable summary for this middle section we needed to go over, how fast and in what direction. We’re going to potentially have two velocities to consider or work with, a scalar component v0, which is the initial velocity at that initial instant of time and then at the end of our problem we’re at a final snapshot that we’re envisioning. I’ll have a final velocity v, just plain v, and that will represent the velocity at time t.
For both of these quantities, the units will be meters per second, and you’ll want to convert to those units if the problem’s not initially given to you in that way. I also want to point out that in some problems it will be v0 will equal v, that that’s okay. In other words, during the scope of our problem if something’s moving along at a constant velocity, this just means that we have no acceleration, and that will be true for some of our problems, and that’s all right. We still can define a v0 and a v. They’ll just happen to have the same value, the same number.
Okay, back to our jetliner problem. Again, I’ve got the same picture drawn here for you, and now I want you to read and interpret this problem and pick out from it or identify what the number values are for v0 and for v. The key to doing this is picture the motion in your mind, and at this instant of time, at the initial snapshot time t0 and at the final time t, what are those instances in this problem and what is the velocity at those instances?
Sometimes I think it helps when you’re trying to pick out or identify what these velocity vectors are to sketch on your drawing a little arrow to represent the velocity here at the start and then at the end. Just have in mind that you’re sketching. You want to represent the direction and then also the relative size. This plane is obviously headed north initially. It’s going faster initially and it slows down eventually, so the velocity vector here at the end would be smaller compared to the initial.
But if you sketch those or add those to your drawing, I think it helps you get a good visual of what the sign should be for these and then it’ll help you picture the value too, which of course you need to get from reading the problem.
After you answer those two, try your hand at this slight variation. It’s the same problem. The only thing I’ve done now is when I set up my origin and my axes I chose a different direction to represent positive y. So with this definition, now answer the same two questions, what quantity should be assigned to the scalar component v0 and the scalar component v? Remember, this scalar component has to have the sign that represents the direction of the velocity at that instant.
These are a little tougher. You may want to use the discussion board to hash it out with your peers because you’ll get 3 points if you get them right and only 1 point if you get them wrong.
Okay, I want to show you a quick little video to get you thinking about what the signs are for these velocity vectors, so for the videos that you’re going to watch, my positive direction is going to be defined to be the right. Based on that definition, when you watch this video, when it starts, count to yourself about one second after it starts, and at that instant tell me what the right sign is for the velocity.
Okay, this motion is going to keep repeating itself. It starts here and then it speeds up moving to the right. Remember, our positive x-axis is defined as pointing to the right. I’m hoping these dots show up okay for you. I haven’t produced this yet to see if the resolution will be good, but this, as it runs this motion, it’s leaving this motion diagram, this spot diagram, to show you that it was speeding up to the right.
So having watched that video from when that object or that blue dot starts at the left, count approximately one second, so it’s somewhere in the middle of its motion, and for question 14, tell me what the sign is for the velocity.
The next little video I’m going to show you asks you again after one second to tell me what the sign of v is. We again are going to use this convention that positive x-axis is to the right.
Okay, the difference, what is the difference between this motion and the last? Again, I’m starting here, and somewhere here in the middle I’m asking you the question, what’s the sign of the velocity at some instant in the middle there? This object you see now is slowing down as opposed to speeding up?
So what’s the sign in the middle of that motion? That’s what you have to answer for question 15.