Visualizing Kinematics

by Jared Rovny

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    00:01 So in these 3 graphs here what I have is a position, a velocity, and an acceleration.

    00:06 In each of them the x axis is still our time, so we're still just tracking what an object is doing as time goes on just like our apple, even if I dropped it or did anything, we can use these 3 graphs to represent, what it is doing as time goes on.

    00:18 So first let's see a simple case. So for example, I just have an object sitting there as we saw.

    00:24 It's just sitting at a particular position say the 1 meter mark and as time goes on, if I'm plotting its position, It would just be a horizontal straight line because it's not moving anywhere.

    00:32 So its position will not be changing. The velocity of this object will be zero as you expect, it might be hard to see but you can see we've actually drawn a horizontal line right on the zero axis of this velocity graph because as time goes on, the velocity is not changing and the velocity was just zero to start with because our object is just sitting still and so again, this is a very simple situation where the acceleration would also be zero because we're not giving our object any dynamics or anything exciting to do.

    00:58 So let's do one different thing now, let's allow it to move. So suppose this object is now moving.

    01:03 First just focus on the position graph. As time goes on, our object is moving to greater and greater positions.

    01:09 So it's moving forward more and more and more in position.

    01:12 That means that its velocity is not zero, it has a positive velocity, but that positive velocity that it has is not a changing velocity so our object has a constant, positive, small velocity which is giving it more and more position in our position graph.

    01:28 So it's very important to notice that our velocity graph is non zero but it is not changing.

    01:33 It has a constant positive velocity. However the acceleration which measures the changes in the velocity of our object is still zero. It still just is not gaining any speed as it moves.

    01:45 Doing one more thing which is giving our object an actual acceleration, which would pick up the acceleration from the zero axis here, but first let's actually look at how we can measure what it's velocity is if our object you move with a constant simple velocity.

    01:59 In our position graph you can see that if I measure a distance from zero up to some point, this would be a displacement because it started at zero and then moved to a new position and so we see that delta x representing a change in position, a change in where it is or a displacement. Meanwhile, the time it took to get to that displacement we call delta t, how long it took for it to get to where it is now. So we have our delta x, our displacement as well, so we have our delta t our change in time, so we know that we could calculate the velocity as delta x over delta t for the average velocity and so we could actually know not only that this object has a positive velocity, as we see in our velocity graph, but we can measure what that velocity should be, where that blue line should be on our velocity graph.

    02:44 And so with our velocity graph we would find that just by taking the displacement delta x from our position graph, and dividing by the time, delta t.

    02:50 An example of this just to make it a little more concrete, suppose that this delta x that you see in our graph is actually 4 meters, suppose an object moved 4 meters, and suppose, it took it 2 seconds to do so, then we could take our equation for the velocity which is the displacement over the time that it took to move that far and then just divide and we see 4 divided by 2 is 2 meters per second.

    03:13 And so on our velocity graph that horizontal blue line will be pinpointed at 2 meters per second, and it would stay at 2 meters per second, so long as our object kept this particular velocity.

    03:22 We can do the same thing when we're finding the acceleration, the average acceleration being the change in velocity over the change in time but notice because the delta v, the change in the velocity is zero, if we do zero divided by a given time we just get zero and so the acceleration graph still shows the acceleration being zero.

    03:40 One last thing to note about this, is that the velocity graph cares about changes in position over changes in time but nothing about the velocity graph tells me where my object is.

    03:50 My object could've started anywhere and ended anywhere as long as it's moving with that same velocity and so with our velocity graph, we have the constant horizontal straight line because it doesn't know or doesn't care about where our object is or where it ended up. It just cares about how it's moving, how it's changing as time goes on and so if I move my position graph up or down, changing the position of my object, it doesn't at all affect my velocity graph, because my velocity graph is just measuring how slope the position graph is which does not depend on where the position graph is.

    04:18 Let's do one last exciting thing which is let's change the acceleration to be non zero, finally.

    04:23 So this would be the case for an object is dropping or an object that has a force acting on it or something like this.

    04:28 If our object is accelerating, the first thing you notice is look at the velocity graph.

    04:32 The velocity graph started at zero but then began to pick up, so you're picking up speed.

    04:37 This would be exaclty the case if you're in a vehicle or something like this starting at rest and then you start accelerating so you're getting faster and faster and faster.

    04:45 As you're getting faster and faster, the velocity is faster and faster.

    04:50 It's increasing, it's going up in our velocity graph.

    04:53 So when that's happening what happens in the positioning graph is that the slope of the position graph, how steep it is getting bigger and bigger and bigger.

    05:02 It's becoming more and more curved upwards. It's moving faster and faster or in other words moving in a given amount of time, further and further and further and so when I introduce a non zero acceleration, as we briefly mentioned before by position graph picks up a curve to it. It actually has a bend on it.

    05:20 So there's some important things to know about what happens when I allow my position graph to curve, first and foremost, I cannot use my average velocity equation to know what my velocity is, because my velocity is changing as time goes on. So at each moment my velocity is something different.

    05:34 So I can't just use my average velocity equation of distance over time because depending in which time interval I pick on my position graph, that value will change as time goes on, meaning that we can't just pinpoint a particular velocity unless we're talking about a specific time on our velocity graph.

    05:51 The second thing is that I could reverse all of these and flip them down, so for example in the penny example where an object is falling from the building, is actually moving downwards and so its acceleration is minus 9.8 meters per second squared.

    06:04 When that's case we do exactly that. We give our acceleration graph a negative value so if the acceleration is negative, then the velocity will be decreasing and decreasing and decreasing getting lower and lower. And similarly our position will be picking up a more and more negative slope to it and sloping down faster and faster as you see here.

    06:23 Now let's do something much more tricky. Suppose instead of me giving you a position graph, and asking you what was the velocity, so if we saw a position graph that was slope positive, we can figure out the velocity by how slope it was or if I gave you position graph that was curve as we saw before, you know that it means a non zero acceleration because the slope is changing but the tricky thing we could do is start with the velocity and ask if I know what a velocity graph looks like, what should the position graph look like? In other words, if I know how fast something is moving, can I figure out how far it moved? So let's look at this velocity graph and notice something, that if I do what is intuitive for finding distances and say that the distance is a velocity times time so suppose for example, you're moving 50 kilometers per hour, that's how fast you're going, and you do that for 2 hours.

    07:10 Well, in this amount of time we've gone a hundred kilometers because you went 50 kilometers per hour for 2 hours but look at what that looks like if we represent it on a graph, what this looks like is this rectangle which has a base of 2 hours, its length at 2 and it has a height the velocity that we're talking about of 50 kilometers per hour, because that's what the height represents on a velocity graph but geometrically notice that what this is you're multiplying a base by a height so the shaded region, in this rectangle, the area of that rectangle is actually representing velocity times time or position, or how far you've moved. And so geometrically speaking again, we can say that this rectangle, the area of the rectangle in the velocity graph shows me how far I moved on my position graph.

    07:56 So in that same 2 hour interval I know that the change in position has to be 100 kilometers because again I moved 50 kilometers per hour for 2 hours.

    08:05 Something very important here which is similar to what I've mentioned before about the position graph being able to move, is that the area of the rectangle tells me how much an object is moving but it does not tell me where the object is.

    08:18 So for example if I say as I did in this example, that you moved 50 kilometers per hour, and you did that for 2 hours, I haven't told you were you started, you could you have started at home, and you've gone 50 kilometers or you could have started at 50 kilometers and gone to a hundred kilometers or something like this and so this 100 kilometers of motion simply tells us that it's motion. So in this position graph I could've move the entire position graph up or down and any of this lines would've been accurate. If you're familiar with calculus this is an integration, where we have an uncertainty in our integration of plus c, plus some constant and so just be aware of that, that a velocity going to a position or similarly an acceleration going to a velocity we will always know changes without knowing actual absolute positions.

    09:01 We can make this one step more complicated. Suppose we have a velocity graph that's not as simple as the ones that I showed before, suppose it's changing, you have one velocity and that velocity changes and then reaches another velocity, could we still find the position, the total displacement of an object that has a velocity graph that looks like this.

    09:19 Fortunately, our geometric arguments still hold.

    09:22 It still turns out that if you want to find the displacement of an object, you can still find the area under this curve that we've represented.

    09:29 The way to do this very practically would be to take each segment of your object and break it up into different areas, areas of simple geometric shapes that you can find.

    09:38 So for example I have here rectangle, and then triangle and then two more rectangles.

    09:42 From the problem, you'll be able to know the dimensions of each of these objects, you will the width and the height of the rectangles, you'll know maybe the base and the height of the triangle depending on what times they gave you and what velocities they gave you.

    09:53 Using the simple geometry you can add up the areas of each of these four objects which I've written as a1 plus a2, etc., the areas of these four objects, and if you add up the areas of the four objects under the velocity graph, then you'll be able to find the displacement of your object and that's a good summary for how graphs work and how to visualize systems and thank you for listening.

    About the Lecture

    The lecture Visualizing Kinematics by Jared Rovny is from the course Translational Motion.

    Included Quiz Questions

    1. The velocity-time graph will be a straight line with a positive slope, and the position-time graph will not be a straight line.
    2. Neither the position nor the velocity graph will be straight lines.
    3. The velocity graph will slope upward, while the position graph will slope downwards.
    4. Velocity will be zero, but the position will be non-zero.
    5. The position graph will cross through the zero axes at least twice.
    1. The velocity is positive and constant, while the acceleration is zero.
    2. The velocity and acceleration are positive and constant.
    3. The velocity and acceleration are positive and changing.
    4. The velocity is positive but changing, while the acceleration is positive and constant.
    5. The velocity is positive and changing, while the acceleration is zero.
    1. This can only be used when the position graph is a straight line.
    2. The formula can never be used for the entire velocity graph.
    3. This can always be used for the entire graph.
    4. This can only be used when the position graph is not a straight line.
    5. This can only be used if the position is not changing.
    1. The velocity and acceleration graphs will be unchanged.
    2. It's impossible to answer without knowing the magnitude of the shift.
    3. Only the velocity graph will change.
    4. Only the acceleration graph will change.
    5. The velocity graph will increase while the acceleration graph slope changes.
    1. The displacement during that time
    2. The average velocity during that time
    3. The average acceleration during that time
    4. The current position of the object after that time interval
    5. The final velocity of the object

    Author of lecture Visualizing Kinematics

     Jared Rovny

    Jared Rovny

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    great explanation
    By Anastasia R. on 21. June 2017 for Visualizing Kinematics

    This lecture has taught me more maths than I learned in one entire year at school. Thank you!