Thursday, May 26, 2022

Position vs. time graphs | One-dimensional motion | Physics | Khan Academy

Let's talk about position versus time graphs these are tricky if you've never seen these these can be really tricky but physicists love these teachers love these throwing lots of tests why do so many people love these things because you could compact a ton of information about the motion of an object into this small little space right here basically .

Specify the entire motion of the object you didn't even have to write an equation or say a bunch of words it's all just right here so these are actually really handy you should know how to deal with these so this graph represents the motion of an object and instead of just saying object let's make a specific let's say it's a turtle a .

Turtle not just any turtle a turtle with a jet pack on this turtle's back and I don't want a sternly worded letter I don't want a bunch of nasty comments let's put a helmet on this turtle who it's a pink helmet she's pretty and now we got turtle safety you always got to use rocket safety all right so let's say this turtle is moving around and this .

Graph represents the motion of this particular turtle the first mistake a lot of people make is they think that well maybe the shape of this graph is the same as the shape the turtle takes through space right so maybe the turtle went forward and then down and then up that's not right in fact turns out that's not even close to figure out what .

This graph actually says let me lay down a horizontal axis here this axis is going to represent the horizontal position so I'm going to label this X and it's going to be measured in meters and I'm doing that because look over here what we're graphing in this case I wrote it as X so this is going to be the horizontal position of the turtle so the .

Horizontal position is what we're actually graphing what that means is if you find the turtle at some point over here at x equals 2 then the graph should represent that the turtle is at x equals 2 by showing the value is 2 so somewhere at like 2 and 1/4 seconds this turtle was at 2 meters and that's what this graphs going to tell you so let's just .

Read this graph and figure out what this particular turtle did if this turtle didn't go forward down and up what did this turtle do well let's start at T equals 0 we'll go up from there and at T equals 0 the value of this graph is 3 and this graph is representing the horizontal position so the value of the .

Graph is giving you the horizontal position so T equals zero the turtle is at 3 meters so let's put her over 3 meters she starts over here 3 meters at t equals 0 now what happens so at t equals 1 seconds same thing we read our graph by going up hit the graph then we go left to figure out where we're at again .

Turtle still at 3 at 2 seconds we come up hit the graph we come over to left to figure out where we're at this turtle still at 3 B that's awkward this turtle didn't even move for the first 2 seconds this turtle is just sitting here so a straight line a horizontal line on a position graph represents no motion whatsoever .

There was no motion this is awkward turtle is probably trying to figure out how to turn on her jetpack should read the instructions sorry all right now what happens turtle at some later time 4 seconds is at negative 5 meters that's all the way back here so between 2 seconds and 4 seconds this turtle rocketed back this way that's also .

Awkward turned on the reverse booster watch on YouTube ha turtle here we go made it all the way back to here and once the turtle do after that point turtle rockets forwards makes it back to 0 at this point and then all the way back to 3 meters of this turtle rock is forward back to 3 meters that's what the turtle did that's what this graph is .

Representing and that's how you read it but there's more than that in here I told you there was a lot of information and there is so one piece of information you can get is the displacement of the turtle and the displacement ah I'm going to represent this with a delta X and remember the displacement is the final position minus the initial position you .

Can find the displacement between between any two times here we're just going to find it for simplicity sake for the total time shown on the graph but it could have found it between 0 and like 4 seconds let's just do 0 to 10 the whole thing so what's the final position the final position would be the position of turtle has at 10 seconds she was at 3 .

Meters at 10 because I read the graph right there – initially because we're considering the total time at zero seconds turtle is also at 3 that means the total displacement was zero and that makes sense because this turtle started to 3 rocketed back to 5 actually started at 3 stood there for a second or two rocketed .

Dr5 rocketed dr3 ended at the same place she started no total displacement what else could we find we could figure out the total distance so the total distance traveled remember distance is the sum of all the path lengths traveled so for this first path length there was no distance traveled that was the awkward part we're not going to talk about that .

Because my hurt feelings then all right so the zero meters plus between three no sorry between 2 seconds and 4 seconds turtle went from 3 to 5 that's a distance traveled of 8 meters and should we make that negative nope distance is always positive we make all these path lengths positive and we add them all up so 8 meters because the turtle went from .

3 all the way back to 5 that's a total distance of 8 meters traveled plus between 4 seconds and 10 seconds the turtle made it from negative 5 meters all the way back to 3 meters that means she traveled another 8 meters that means in total the total distance traveled was 16 meters for the whole trip again you could have found this for two points any .

Two points on here all right what else can you figure out you can figure out the say average velocity sometimes people represent that with a bar sometimes they just say the AV jeeps AVG that is sloppy AVG what does this mean remember average velocity is the displacement per time and let's find the total right so we're finding the total .

Values here so the total average velocity well I need the total displacement already found that total displacement was zero for the whole trip so zero meters divided by it doesn't really matter now but 10 seconds was the time it took for that entire displacement power not meters 10 seconds so this equals zero there's no .

Total average velocity the average velocity for the entire trip was zero because the turtle had no total displacement how about average speed so the average speed I'm just going to write it as average speed maybe you'll see it as an S with a bar maybe an S with an AV gee I don't know physicists use all kinds of letters you don't know .

What you're going to get but the average speed is defined to be the distance per time and again let's try to find the total average speed the whole 10 seconds that's not too bad cuz I already found the total distance that was 16 meters so 16 meters divided by the total time it took 10 seconds for that entire trip this turtle she was going 1.6 meters per .

Second on average that was her average speed probably would have been a little higher if she didn't have that technical difficulty here at the beginning all right we can figure out more than this though we can figure out the instantaneous velocity maybe you'll see it as V I NS t maybe you just see it as V because this is usually what we're .

Talking about we're talking about velocity I'm talking about the instantaneous value a lot what is this here's the key idea in fact this is maybe the most important idea of this whole video to find the instantaneous velocity when given a position versus time graph you look at the slope because it turns out the slope of a position .

Versus time graph is the velocity in that direction so since we had a horizontal position graph versus time this slope is going to give us the velocity in the X direction and not only that if we find the average slope we get the average velocity and if we find the instantaneous slope we're going to get the instantaneous velocity so how do I .

Do that how I find the instantaneous slope well in general if you got a curved graph you're gonna have to use calculus but we're in luck here because look at these lines they're all straight and what that means is that the average slope between any two points on one of these lines is going to equal the instantaneous slope at any point on the .

Line so let's make this specific let's say we want to find the instantaneous velocity at shoot I don't know 3 seconds pick any point three seconds how do we do that well we got to figure out what we mean by instantaneous velocity we mean the velocity at three seconds slope here but I got to go to the graph so I take my three I go down to the graph I .

Want to know what the instantaneous slope was at that point right there let me draw on top of this thing here I want to know what the slope was right there how do I do that well I told you the key is that the average slope between any two points on this line so I can pick these two if I want the average slope between these two points is going .

To equal the instantaneous slope at any point because look at this slope isn't changing slopes the same the whole way and if you take the average of a bunch of quantities they're exactly the same you're just going to get the same value as any one of these quantities that was a complicated way of saying if you took the average of 8 & 8 & 8 & 8 what are .

You get the average value of those is 8 which is the same as any one of these values so if you ever have a graph that's a straight line you're in luck you'll need calculus you find the average velocity by taking or sorry you can find the instantaneous slope at any point by taking the average velocity velocity between any two points .

I'm picking these two why these two because they're convenient look it I know exactly what they're at that's 3 & 2 and this one's negative 5 & 4 you might wonder why why is this true why's the velocity equal to the slope well remember from math class slope was the rise over the run and you might have seen that as okay why – this is math .

Class here – y1 over x2 minus x1 but you saw it like that because in that class typically the vertical axis was always Y and the horizontal axis was always X this is physics opera axis isn't X horizontal axis is T and our vertical axis is what we're calling X so for physics class the slope of this graph particularly the rise in this case is .

This axis so it's going to be x2 minus x1 over the run well that's going to be t2 minus t1 all right so how do we do this well this is point two this is point how do you know this isn't too and that's one the point further in time is the one you choose as the second point so at four seconds and negative five .

Meters that's our point – all right so X – that would be negative five because they just read my graph at point 2 that's negative five so I've got negative five meters minus x1 that's this don't make X 1/4 that's a time that's not a disturb that's not a position so 0.1 the horizontal position was 3 so positive 3 but the negatives .

Here because the negatives in the formula and then divided by a time – that was 4 seconds and minus t1 was let's see two seconds and if you set it if we solve this thing negative 5 and negative 3 is negative 8 divided by 2 seconds hoops can't forget my units I look at that I got negative 4 meters per second that was the instantaneous .

Velocity at three seconds negative 4 meters per second negative because the turtle was going backwards number that was the awkward she turned on the reverse booster set a floor or booster negative and 4 because looketh going 4 meters every second made it eight meters in two seconds that means she was going 4 meters per second on average and since .

It's a straight line that was the rate she was going at any moment beautiful all right that would have been if if the follow-up question is what is it at two point four seconds don't get concerned look it's the same everywhere it'd be the same answer negative 4 meters per second for this whole line what else can we figure .

Out one last thing let's say you were asked what's the instantaneous speed at a point some write that as s/insp instantaneous speed or just ask this usually won't mean by speed equals well average about or sorry absolute value of the instantaneous velocity so now here I've got to make an assumption this is going to get a little subtle if all .

We're giving off sorry if all we're given is a horizontal position graph we don't really know about the vertical position this turtle could have gone back and forth where the turtle could have been like flying upward as she went back and forth and if the horizontal location was the same the whole way this would have looked exactly the same .

Regardless of whether the turtle had any vertical motion at all so we got to be careful because the speed is the magnitude of the tomé velocity this is just the velocity in the X direction so we're going to make an assumption I'm going to soon this turtle was just moving horizontally this children having a vertical motion she's not ready for .

That yet all right so how do you get this this speed is just the absolute value the magnitude of the instantaneous velocity and if this is the only component of velocity then I can just figure this out pretty easy by saying that he hat I got to give you time ha ha makes no sense to say instantaneous .

Speed I gotta say instantaneous speed at given moment because the instantaneous speed here was zero the instantaneous speed at this point would have been what well it would have been the absolute value of this so it'd have been positive for scuse me positive for m/s that would have been the instantaneous speed at three seconds or any time between two to .

Four seconds really whoo that was a lot I told you there's a lot in there so recapping really quick the value of a horizontal position versus time graph gives you the horizontal position surprise surprise the slope of a horizontal position versus time graph gives you the velocity in the X direction the average slope gives you .

The average velocity the instantaneous slope gives you the instantaneous velocity and if it's a straight line with no curvature he's going to be the same on any given line they weren't the same here you're like what hold on these aren't the same well that's because I averaged over this whole thing right here I took the average velocity over .

The whole time and this slope is changing so what I really got was the average of all of these and that's why these weren't equal but if I just if I can train myself to just the average value along a line that doesn't change its slope that will equal the instantaneous slope at any point and the instantaneous speed is the magnitude of .

The instantaneous velocity assuming the only on motion one erection


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