hey everyone welcome to the second video on linear motion we'll pick up where we left off in the last video and learn a few more things in this video we'll learn the last two kinematic equations that we'll need for linear motion and we'll talk about equations for horizontal and vertical motion we'll also learn the difference between scalars and vectors and we'll talk about physics problems involving average speed first up let's cover the last two kinematic equations that we'll need when we're studying the linear motion of objects and see how we can apply them to both horizontal and vertical motion just as a quick reminder here are the kinematic equations for position displacement velocity and acceleration that we learned in the last video these equations are what we might call the definitions of position velocity and acceleration the next equations that we'll learn are really useful and they combine all three of these so this first equation includes the variables for time position velocity and acceleration and for an object that's accelerating this equation helps us find the object's position at any point in time by using the initial position the initial velocity the acceleration and how long it's been accelerating for let's take a closer look so in this equation XF stands for final position x i is the initial position VI is the initial velocity T is the time and a is acceleration something to note here which we covered in one of the basics videos is that when we have two variables next to each other with nothing in between them it means we have to multiply them together so here we have initial velocity times time and we have one half times acceleration times time squared another important thing to point out is that this equation only has the variable t for time it doesn't have a delta T for change in time and there's no initial time and final time the thing to remember is this for equations that only have a t by itself the T really means delta T or change in time or we could also think of it this way we assume the initial time is just zero and T is equal to the final time which is the time point that we care about the time point we're plugging in to find the object's position and here are the SI units that we use for this equation time is in seconds position is in meters velocity is meters per second and acceleration is meters per second squared all right now the last kinematic equation that we'll learn for linear motion is this one this equation relates the position velocity and acceleration of an object but it doesn't include the variable for time which is pretty interesting because time is a fundamental part of kinematics this equation is actually made by combining and rearranging a few other equations in order to get rid of the variable for time which makes it really useful for certain problems where we don't know anything about time so for an object that is accelerating like this car this equation will help us find the final velocity of the object based on its initial velocity its acceleration and its change in position this looks very similar to the equation we learned for acceleration if we had rearranged the variables like we mentioned in the last video so with the equation on the left we're using the change in time which is the part in parentheses and with this new equation on the right we're using the change in position to put that into words the left equation can tell us the car's velocity based on how much time the car has been accelerating for in the right equation can tell us the car's velocity based on how much distance the car has been accelerating over so let's take a closer look at the variables in this new equation VF stands for final velocity VI is initial velocity a is acceleration XF is final position and x i is initial position and don't forget multiplication is implied in these two spots as a whole this equation says that for an object the final velocity squared is equal to the initial velocity squared plus two times the acceleration times the change in position or the final position minus the initial position and here are the SI units for the variables in this equation awesome so now we have a nice list of kinematic equations that we can use to solve problems which we'll work through more in later videos and don't worry if this seems like a lot here you don't have to memorize it all of these are on your equation sheet and the variables are defined there so you'll know what they mean now before we wrap up with kinematic equations there's one more thing we need to cover and that's the variables that we use for horizontal and vertical motion so we're learning about linear motion or motion along a straight line and mostly we've talked about a car driving along a road which would be horizontal motion but we can also have vertical motion like if we dropped a ball from some height and let it fall down the car and the ball are both examples of linear motion but they're in different directions in physics we typically use x to represent horizontal motion and we use y to represent vertical motion so just like we've been using X to represent horizontal position we're going to use y to represent vertical position or the height of an object and sometimes we'll use X and Y as subscripts to label the direction of something we can use V with X as a subscript to mean velocity in the horizontal Direction or V with Y as a subscript to mean velocity in the vertical Direction likewise ax would mean horizontal acceleration and a y would mean vertical acceleration now when you look at it organized this way at least for me it kind of feels like position should have its own variable like P and then we would use X and Y as subscripts just like velocity and acceleration do well the variable p is already taken but you'll actually find in some places that people use the variable s to represent position but in most cases I've seen everyone use X and Y to represent position so that's what we're going to do in this course the takeaway here is that when they're the actual variables X and Y mean horizontal and vertical position but when they're subscripts next to a variable X and Y are telling us the direction of that variable like velocity or acceleration so what does this mean for our list of kinematic equations that we've put together well it means we actually know twice as many equations as before without even learning them every equation can be applied to either horizontal or vertical motion just by using X or Y the equation we learned for displacement uses X for horizontal displacement but if we use y instead of x now we have an equation for vertical displacement here's our equation for average velocity that uses X and again if we use y instead of X we have an equation for vertical average velocity notice how we've added X and Y as subscripts for the average velocity variables to label them as average velocity in the horizontal or vertical Direction and we can do the same thing for our equation for acceleration here all the variables for velocity and acceleration get their own subscript X or Y to label the direction however the variable t for time does not get an X or a y subscript because time applies to all motion time can't be horizontal or vertical and finally here are the last two kinematic equations that we learned and again each one has a version for horizontal or vertical motion just by using either X or Y so to wrap up this section here's what I want all of you to take away from this different physics equations are going to show up in this course in your class in other online videos maybe on a website where you do your homework or wherever and all of those equations might look just slightly different from each other they might have different subscripts or they might just be rearranged and it can definitely be annoying especially when you're learning these for the first time but don't worry about all of that all you need to know is what each variable means and what each subscript means that way you'll understand how to use any equation even one that you haven't seen before so just to reiterate the variables in an equation are the larger letters and the subscripts are the smaller letters at the bottom right of the variable we can think of a subscript as a label and a variable can also have more than one subscript so for the kinematic equations here are the variables that we'll use T is time as variables X and Y mean position V means velocity and a means acceleration and for the subscripts I means initial or instead you might see a zero which means time zero and it's also sometimes called not I and 0 both mean the same thing f means final and if there's no subscript it might also mean final you would see variables with the subscripts I and F in an equation or instead you would see variables with a subscript 0 and then one with no subscript these are sort of paired together then X as a subscript means the horizontal Direction and Y as a subscript means the vertical Direction and last it's not a variable or a subscript but remember that this little triangle is Delta and it means the final value minus the initial value or the change in that value and again this might seem like a lot to remember but don't worry the equation sheet for this course will have everything on it this video is meant to introduce these equations but we'll get a lot more comfortable with them as we start doing practice problems so next up let's talk about scalars and vectors a scalar is a quantity that only tells you the magnitude and a vector is a quantity that tells you both the magnitude and the direction so what does that mean well scalars are what we're mostly used to for example six miles is a scalar quantity and 12 meters per second is a scalar quantity on the other hand six miles north would be a vector or 12 meters per second to the right would also be a vector that's because they tell us the magnitude like six miles as well as a direction like North if you take a scalar quantity and add a direction to it you turn it into a vector also we've mentioned the words distance and displacement the difference between them is that distance is a scalar and displacement is a vector and it turns out this is the difference between speed and velocity speed equals distance over time which makes it a scalar while velocity equals displacement over time so it's a vector let's take a look at the difference between distance and displacement here's our friend Mike if Mike starts here and he travels 5 meters where is he now well it's not possible for us to know because we weren't given a Direction we were given a distance which is a scalar now instead say we have a compass and we're told Mike travels 5 meters West now we know exactly where he is here we were given a displacement which is a vector and it includes both the magnitude and the direction think of it as taking a distance value and adding on a Direction here's another important distinction between distance and displacement here's Mike again let's say that Mike walks in a full circle with a circumference of 15 meters and ends up right where he started what is his distance traveled and what is his total displacement well his distance traveled which is the actual path that he covered while he was walking is 15 meters that's the circumference of his Circle or the length of his path however his total displacement during this journey is zero why is that well remember our equation for displacement is the difference between the final position and the initial position and for this journey Mike started and ended in the same place so his final position and his initial position are the same and that gives us that the displacement is zero let's look at this another way let's say you're driving from your home to your school there's different paths you could take but let's say you drove to school like this the distance you traveled let's say is five miles what would be your displacement well the displacement is the direct difference between your initial and final points and we can think of it as an arrow going from your starting point to your ending point so here our displacement turns out to be three miles Northwest displacement is a vector so it has both a magnitude three miles and a direction Northwest so here's a question for you every car has something called an odometer which measures the number of miles driven in your car if we check the odometer at home and then checked it again when we got to school with the odometer increase by 5 miles or would it increase by three miles so does the odometer track our distance or our displacement the answer is the odometer would go up by five miles because it measures distance it measures the actual path that the car takes so that's another way to think about the difference between distance and displacement next we can actually use the same examples to compare speed and velocity again Mike walks in a full circle with a circumference of 15 meters but now we're given that it takes him 10 seconds what is his average speed and his average velocity well speed is equal to distance over time and his distance traveled is 15 meters and it takes them 10 seconds so his average speed is 15 divided by 10 or 1.5 meters per second but velocity uses displacement not distance we learned last time that his displacement is zero meters since there's no difference between his initial and final position so Mike's average velocity is 0 meters divided by 10 seconds or 0 meters per second that seems kind of strange because we know Mike was moving as he walked around the circle but this is how we Define average velocity and in this case it's zero what about our driving example we already found the distance and displacement of our car and let's say the trip takes us 12 minutes which equals 0.2 hours so what would be our average speed and our average velocity for this trip well speed is distance over time which would be 5 miles divided by 0.2 hours and we get 25 miles per hour so that's our average speed next velocity is displacement over time so to find the magnitude we would do three miles divided by 0.2 hours which gives us 15 miles per hour but velocity is a vector so we need to include the direction so our average velocity would be 15 miles per hour Northwest it has the same direction as our displacement and similar to last time every car has what we call a speedometer that's the thing that tells you how fast you're going now does the speedometer tell us the speed or the velocity of the car well the name kind of gives it away but a speedometer tells us the car's speed the speedometer only knows we're driving forward but not which direction we're traveling however if your car has a built-in Compass you could think of combining your speed and your compass Direction together to give you your car's velocity which is a vector alright so the last thing we're going to cover in this video is physics problems involving average speed so here's our equation for average speed average speed equals the total distance traveled divided by the total time it took to travel that distance I emphasize total because it's an important part of finding average speed for example if we said that this car traveled 30 miles and it took two hours then the average speed of the car for that period is 30 miles divided by two hours or 15 miles per hour seems pretty straightforward right but sometimes physics problems can trip us up because the information we get isn't this easy and we might forget that average speed is defined as the total distance over the total time if the problem wants us to find average speed this is how we have to calculate it take a look at this example a car drives 150 miles at 50 miles per hour then drives for another 150 miles at 75 miles per hour what is the average speed of the car for the whole trip so what are the important pieces of information in this word problem we're given some numbers the car drives 150 miles at 50 miles per hour and 150 miles at 75 miles per hour and we're being asked to find the average speed so we see the word average we know what average means we know how to take a group of numbers and find the average right so to find average speed can we just find the average of 50 miles per hour and 75 miles per hour that would be 50 miles per hour plus 75 miles per hour divided by 2. which gives us 62.5 miles per hour that seems logical right well this is actually not the average speed remember this is how we Define it the average speed equals the total distance divided by the total time so we know the total distance 150 miles plus 150 miles gives us 300 miles but we don't know the total time how can we find it well we know the distance and the speed of the car for each part of this trip so what we can do is find the amount of time the car traveled for each part of the trip and then add them together to get the total time so speed equals distance divided by time but we want to rearrange this equation so we can find time if we multiply both sides by time we get speed times time equals distance then if we divide both sides by speed we get time equals distance divided by speed quick side note we just rearranged our equation so time is by itself on one side and then next we're going to plug in numbers to solve for time we could have also plugged in numbers first and then rearrange the equation for time I just chose to do it this way okay so for the first part of our trip we have 150 miles divided by 50 miles per hour which equals three hours for the second part of the trip we have 150 miles divided by 75 miles per hour which equals 2 hours so the total time for the entire trip is three plus two or five hours now we can calculate the average speed of the car for the entire trip average speed equals the total distance 300 miles divided by the total time or five hours which equals 60 miles per hour so that is the actual average speed that this problem is asking us to find so why is this answer correct and why is it different than the one we got before let's explore this for a minute you don't need to remember this but I think it'd be nice to clear things up so this first method of finding average speed feels right doesn't it 62.5 is halfway between 50 and 75 which is how we're used to finding the average of two numbers but the correct average speed 60 is less than that it's closer to 50 than it is to 75. so why is that the car even drove the exact same distance at each of these two speeds so it feels like each speed is equally important like if you imagine to balance each speed would have equal weight now if instead the car had driven 50 miles per hour for a thousand miles and 75 miles per hour for only one mile then it would feel like the 50 miles per hour would hold more weight and the average should be closer to 50. but the distances are actually the same however the times are not the same we found that the car was driving 50 miles per hour for three hours and 75 miles per hour for only two hours so since the car was driving at 50 miles per hour for a longer time the 50 has more weight to it so the average speed is closer to 50 than 75. so really when we say average speed in physics you could think of it as the time weighted average speed as opposed to the distance weighted average speed when an object like a car travels at different speeds over its Journey we're averaging the speeds based on their times and not on their distances anyway you can forget about that the nice thing is all you need is this equation if a problem asks you to find the average speed just divide the total distance by the total time and you'll be good to go all right let's do a recap of what we learned in this video first off we learned the last two kinematic equations for a linear motion which both apply to an object that's accelerating next we learned that we can describe horizontal motion using the variable or the subscript X and we describe vertical motion using the variable or subscript y and that let us use each of our kinematic equations for either horizontal or vertical motion by using either X or Y we also learn that all we need to know is what each variable and each subscript means and then we don't need to memorize any equations next we learn the difference between scalars and vectors where scalars have only a magnitude but vectors have both a magnitude and a Direction and we covered a few examples showing the difference between distance and displacement and speed and velocity and finally we talked about how average speed could be thought of as the time weighted average speed but when we're solving problems like these all we need to do is simply use this equation for average speed with the total distance and the total time all right thanks everyone for watching and I'll see you in the next video