in this video we're going to go over motion graphs we're going to talk about position time graphs velocity time graphs and acceleration time graphs now there's two concepts you need to be familiar with the slope and the area the slope is associated with division the area is associated with multiplication perhaps in algebra you've seen the slope represented by the letter m the slope is calculated by dividing the change in y by the change in x so when you're calculating slope you're using division now let's say if we want to find the area of this rectangular segment let's call it segment two to find the area we need to multiply the length by the width in this case we're multiplying the y values by the x values so this would be the change in x that would be like the width of that rectangle and this is the change in y which would be the height of the rectangle if we multiply those two that will give us area so associate area with multiplication but slope with division in algebra you've seen this equation for slope it's the change in y which is y2 minus y1 divided by the change in x which is x2 minus x1 and you've seen this equation for area areas length times width and then with a rectangle so make sure you understand those concepts now let's consider three common time graphs that you're going to encounter the first one is the position time graph typically it's going to be x versus t so when you see that you're dealing with motion along the x axis it could be y versus t if you're dealing with motion along the y axis v versus t is the velocity time graph a versus t is the acceleration time graph now what do you think the slope of a position time graph represents remember slope is the change in y over the change in x you're dealing with division so if we were to divide the position or the change position by the change in time what will we get the change in position divided by the change in time the change of position is displacement if you divide displacement let me just put d for displacement instead of writing everything out but if you were to divide displacement by time you're going to get the velocity so the slope of a position time graph represents the velocity or the instantaneous velocity at that instant if you calculate the slope the tan at let's say like i add a tangent you're going to get an instantaneous velocity if you calculate the slope using two points you're going to get an average velocity by the way calculating the slope at one point is like finding the slope of the tangent line that will give you the instantaneous velocity calculating the slope using two points which is the slope of the secant line that will give you the average velocity nevertheless the slope of the position time graph gives you velocity so make sure you understand that now you might be wondering what does the area of a position time graph tells us the area doesn't tell us anything remember area is associated with multiplication if we were to multiply the y-axis by the x-axis in this case the change in x by the change in t we would get meters times seconds which it really doesn't help us in physics so the area for position time graph is not really helpful now what about the velocity time graph what does the slope tell us well in order to find out the slope of a velocity time graph we need to use division if we were to divide v by t if we were to take the change in velocity and divided by the change in time what will we get velocity typically has units meters per second and time is usually in seconds when you divide these two you get the units meters per second squared what variable or term is associated with the units meters per second squared from physics you should likely see it as acceleration the slope of a velocity time graph is the acceleration so that's what you need to know when dealing with a vt graph now what about area what does the area of a velocity time graph tell you so once again when you think of area think of multiplication if we were to multiply the y-axis by the x-axis if we were to multiply v by t what will we get well let's look at the units velocity is usually in meters or per second or meters over seconds time is usually in seconds when you multiply these two the unit seconds cancels and so we get the unit meters meters is the unit for distance position displacement but it turns out that when you multiply velocity by time you get specifically displacement so i'm going to put d for displacement speed multiplied by time is distance but velocity times time is displacement so the area of a velocity time graph is specifically displacement so when dealing with a velocity time graph the acceleration i mean the slope and the area is important the slope is the acceleration the area is the displacement displacement is the final position minus the initial position if you move this term to the other side you're going to get this familiar equation x final is equal to x initial plus v t let me erase that because i'm going to need the space on the right side so hopefully you're taking notes or writing these things down because we're going to use this later in the video now let's move on to the acceleration time graph what does the slope of an at graph represent well let's divide y by x so on the y axis we have acceleration on the x axis we have time so what is the rate of change of acceleration now for most physics classes that you're going to encounter you're not going to have to worry about it this is not going to be applicable to the everyday common situation but some physics course they do have a value for this and this represents jerk or joe you can look this up on wikipedia you may see in some textbooks but that's the slope of an acceleration time graph on a physics test if you're asked the question what is the slope of an acceleration time graph and you don't see jerk or drop you might have to go with nothing because maybe your teacher didn't cover that topic because for most physics course you won't see this but in the event that you do see those terms that's what it is now what about the area of an acceleration time graph by the way the units for this would be meters per second squared over seconds so it would be meters per second cube so that tells you dealing with the rate of change of acceleration by the way the rate of change of position is velocity the rate of change of velocity is acceleration and the rate of change of acceleration as we just mentioned is jerk or dro now for the area we're multiplying y by x in this case acceleration by time so what is acceleration multiplied by time acceleration has the units meters per second squared if we multiply that by seconds we get the unit meters over seconds which represents the unit for the velocity so a times t gives you the change in velocity if you replace delta v with v final minus v initial you get this familiar equation v final is equal to v initial plus a t so the area of an acceleration time graph is the change in velocity now let's qualify that statement so let me change this graph real quick so let's say we have a curve that looks like this let's say that's our position time graph and let's put some numbers let's say this is 1 this is 3 and this is five now if we draw a line at three a line that touches the curve at one point my lines are terrible let's do this one more time i will make the best of that but a line that touches a curve at exactly one point and that line is known as the tangent line the slope of a tangent line gives you the instantaneous velocity if you're dealing with a position time graphs let me say that one more time the slope of the tangent line of the position time graph gives you instantaneous velocity that is the velocity instantly when t is equal to three seconds now let's say if we have two points at one and five and let's draw a line connecting those two points a line that touches the curve at two points that line is a secant line the slope of a secant line gives you the average velocity the slope of the tangent line gives you the instantaneous velocity to calculate the instantaneous velocity that's a little difficult because it's hard to find a slope at one point now you could use calculus you could find the derivative and that can give you the slope of the tangent line which is the instantaneous velocity the slope of the secant line you could use algebra to get that answer you could use uh y2 over y1 divided by x2 minus x1 so it's easy to find the slope of the secant line which is the average velocity now you can approximate the slope of the tangent line using the slope of the secant line so if we want to get a good estimate of the slope at exactly t equals 3 if we know what the position is at let's say 2.9 and 3.1 we can calculate the slope of the secant line of those two points 2.9 and 3.1 to approximate the slope of the tangent line at 3. the closer those two points gets a three the more accurate the secant line becomes to the slope of the tangent line so as those two points get closer and closer to three the slope of the secant line approximates the slope of the tangent line so that's how you can find the slope of the tangent line using this formula so if you were to use values like 2.99 and 3.01 it's going to be a very accurate estimate of the slope of the tangent line so let's put this all together when dealing with a position time graph the slope of the tangent line gives you the instantaneous velocity the slope of the secant line gives you the average velocity when dealing with an acceleration time graph the area doesn't give you the instantaneous velocity nor does it give you the average velocity but it gives you the change in velocity that is v final minus v initial so just make sure you see that distinction and remember the slope of a position time graph is velocity the slope of a velocity time graph is acceleration the slope of an acceleration time graph is jerk or jolt which is not commonly used and the area of a velocity time graph is displacement the area of an acceleration time graph is velocity those are things you have to know if you want to answer questions with these time graphs now i like to make a distinction between two similar time graphs a position time graph and a distance time graph in physics position time graphs you'll typically see them as x versus t for a distance time graph you'll see them as d versus t now what you need to know is that velocity which i'll just put v velocity is displacement over time speed is equal to the distance divided by the time so here we're dealing with division so think of division as slope the slope of a position time graph is velocity but the slope of a distance time graph is speed because remember the slope is d over t distance over time which is speed so that's the difference between a distance time graph and the position time graph the position time graph can give you velocity if you calculate its slope but a distance time graph let me make sure i said that correctly a position time graph can give you the velocity if you calculate the slope but the distance time graph can give you the speed if you calculate the slope so remember velocity is a vector and speed is a scalar quantity velocity can be positive or negative but speed is always positive so make sure you see the difference between the two if you're dealing with a position time graph versus a distance time graph now let's take some more notes so we said that velocity is the rate of change of position therefore as the position increases the velocity is positive when the position is decreasing velocity is negative so if x is going up that means that the particle or the object is moving to the right along the x-axis if x is going down that means it's moving to the left so anytime velocity is positive when you're dealing with an x versus t graph that means the particle's moving to the right if velocity is negative it's moving to the left if the position is constant that means the velocity is zero now when the velocity is zero it could be that the object is at rest or it could be that the object is changing direction so it really depends on the shape of the graph so for instance let's say if you have a position time graph that looks like this so notice that it's horizontal for quite some time at that moment the particles at rest but it could change instantly let's say if it looks like this well let's do it like this actually so notice that it's horizontal for a very brief moment the tangent line which is the slope at that one point the slope of that tangent line is zero because the line is horizontal and so at that instant it's at rest but it's changing direction so here the position is increasing that means the particle's moving to the right because the slope is positive does velocity is positive here it's going down that means the slope is negative which means velocity is negative so it's moving to the left so it was moving to the right and now it's moving to the left so at the top at that peak where the slope is zero it's at rest for a very very short time or more specifically it's changing direction going from right to left so when velocity is zero the particle could be at rest or the particle could be change in direction so keep that in mind now we said that velocity is the rate of change of position acceleration is the rate of change of velocity so whenever the acceleration is positive that means that the velocity is increasing when the acceleration is negative the velocity is decreasing if the acceleration is zero that means that the velocity is constant now speed is the absolute value of velocity so if the velocity is positive five meters per second that means that the speed is positive five if the velocity is negative four meters per second that means that the speed is positive four meters per second so if a particle's moving to the left at four meters per second you would say that the speed of the particle is simply four meters per second speed is always positive now we need to talk about when an object is speeding up versus when it's slowing down how can you determine when it's speeding up and when it's slowing down here's a quick and simple way to get the answer a particle is speeding up when the acceleration and velocity have the same sign either they're both positive or both negative in this situation the velocity is increasing it's becoming more positive because the acceleration is positive here even though the velocity is negative it's becoming more negative and so if you get a larger negative the speed which is the absolute value of velocity that's becoming more positive so in both cases it's speeding up when the signs of acceleration and velocity are different where one is positive and the other is negative that's when the particle or the object is slowing down so let's think about this conceptually here when the acceleration is positive that means velocity is increasing velocity is becoming more positive which means speed has become more positive so speed is increasing the acceleration here is negative and the velocity is negative because the acceleration is negative the velocity is becoming more negative so think of it as going from negative five to negative eight but speed being the absolute value of velocity is going from five to eight so speed is increasing in that case now for this situation the velocity is negative but the acceleration is positive which means that the velocity is becoming less negative so to illustrate this let's say if the velocity was negative five acceleration is making it less negative more positive so it would become like negative 2 due to a positive acceleration the velocity is increasing when acceleration is positive so negative 2 is higher than negative 5 on a number line but if you take the absolute value of it you can see why the speed is decreasing going from five to two s is going down now a quick illustration for the last one where acceleration is negative but velocity is positive the velocity is becoming less positive so let's say if it was eight with a negative acceleration it can go down to four speed being the absolute value of velocity will be the same going from eight to four so the speed will be decreasing thus any time an object is slowing down the acceleration and velocity have opposite signs when it's speeding up the acceleration and the velocity have the same sign so that's a quick and simple way to determine if an object is speeding up or if it's slowing down now let's focus on the three linear shapes of a position time graph now these linear shapes exist for any graph so you can have a straight line going up you could have a straight line going in a horizontal direction or a straight line going down those are the three linear shapes that you're going to be dealing with now because these shapes are linear the slope is constants and for position time graph velocity is the slope so for these three situations velocity is constant and when velocity is constant what can you say about acceleration anytime velocity is constant acceleration is zero so the acceleration is zero for each of these three position time graphs now for the first one the position is increasing any time the position is increasing the velocity is positive for the second one the position is not increasing its constant so the velocity is at zero which means it could be at rest or it may be changing direction but for this particular shape here it's at rest it's not changing direction it's not going up and then down but when v is zero if you don't have the graph just know that it could be at rest or it could be changing direction here the position is decreasing whenever the position decreases the velocity is negative so when dealing with a position time graph if you have these three shapes just no acceleration is zero if it's going up velocity is positive if it's going down velocity is negative if it's horizontal velocity is zero so in this side it's moving to the right along the x-axis here it's moving to the left and for this particular picture specifically this one it's at rest now let's consider the next four fundamental shapes that you'll see with a time graph so now we don't have the three linear shapes that we did before but we have four parabolic shapes so because the position time graph is not linear the velocity is not constant therefore we have an acceleration but before we go into acceleration let's talk about velocity so for the first one is the velocity positive or negative well the position is increasing because we're going up along the y axis so therefore velocity is positive because the slope is positive here the position function is decreasing so therefore velocity is negative for this particular shape the position function is still decreasing because we're going down so velocity is negative as well but here we're increasing so x is going up therefore velocity is positive so that's the velocity for each of those four situations now what about acceleration in calculus if you've taken it before this shape is concave down this shape is known as concave up when dealing with a position time graph if you have a concave down shape the acceleration is negative so down for negative and up for positive so for now go ahead and write that down so notice that these two combined form a concave down shape i put them like that together so you can easily tell that the acceleration will be negative these two shapes combined form a concave up shape this is the first half of it and that's the second half so for the first two shapes acceleration is negative and for the last two acceleration is positive now let's look at this from another perspective we know that the slope tells us the velocity but the way that the slope changes tells us the acceleration velocity is the rate of change of position but acceleration is the rate of change of velocity so if we analyze how the slope changes we can get an idea of the acceleration for this graph so at this point the slope appears to be approximately one this is a slope of one when it rises at a 45 degree angle when it's horizontal the slope is zero if it goes down at a 45 degree angle the slope is negative one so at this point the graph appears to be rising at a 45 degree angle so the slope is approximately one here but then it becomes almost horizontal where the slope is approximately zero so the slope is going from one to zero even though the velocity is positive because x is increasing we're going up the slope is decreasing it's going from one to zero therefore we can see why the acceleration is negative remember the slope represents the velocity so if the velocity is going from one to zero it's decelerating that's why we can say the acceleration is negative so now let's analyze the slope for the second one so here it appears horizontal and at this point it looks like it's going down at a 45 degree angle so the slope or the velocity is going from zero to negative one so it's still decreasing negative one is less than zero so we can see why the acceleration is negative let me keep these numbers here now at this point the slope appears to be negative one it's going down at a 45 degree angle but it's becoming horizontal where the slope is zero so going from negative one to zero the slope or the velocity is increasing thus we can see why the acceleration is positive and here it's clear to tell the slope appears to be zero and here it's going to one so from zero to one the velocity is increasing which means the acceleration is positive so anytime the acceleration is negative the velocity is decreasing when the acceleration is positive the velocity and the slope is increasing now the last thing that we need to talk about is if it's speeding up or slowing down so looking at the first graph on the left would you say it's speeding up or slowing down so if we look at the signs for velocity and acceleration they're different so we can say that it is slowing down remember speed is the absolute value of velocity if you go from one to zero you're slowing down here the signs are the same so it's going to be speeding up this time velocity is zero speed being the absolute value of velocity is not negative one but positive one so in this case the speed is going from zero to plus one it's speeding up here the speed is going from one to zero so we can see why it's slowing down now for the next case the signs are opposite so therefore it's going to be slowing down so if velocity is negative one speed is positive one so going from one to zero we could see why it's slowing down and for the last case both velocity and acceleration have the same sign so it's going to be speeding up these numbers none of them are negative so speed is going to be the same it's going to be 0 to 1. and so in that case it's speeding up as well so now you can answer almost every question when dealing with position time graphs you know how to determine if the velocity is positive or negative if it's increasing or decreasing thus you know how to determine the sign of acceleration and whether if it's speeding up or slowing down you