WEBVTT physics/high-school-physics/selhorst-jones 00:00:00.000 --> 00:00:05.000 Hi, welcome back to educator.com. Today we are going to be talking about multi-dimensional kinematics. 00:00:05.000 --> 00:00:09.000 What happens when we are moving in more than just one dimension! 00:00:09.000 --> 00:00:11.000 So, previously we dealt with everything as if we were only one-dimensional, always just one dimension. 00:00:11.000 --> 00:00:17.000 Otherwise, everything is a scalar, a single number on its own. 00:00:17.000 --> 00:00:22.000 But, now we are going to see some of the concepts we were dealing with before, actually are vectors in hiding. 00:00:22.000 --> 00:00:30.000 All the vectors, all the ones we were talking about, are getting from one place to another place, because a place can be many dimensions. 00:00:30.000 --> 00:00:31.000 We are used to living in a three-dimensional world. 00:00:31.000 --> 00:00:39.000 You do not just go to the store on a direct path, not everything is always going to be a straight line, if you look at two paths or more paths. 00:00:39.000 --> 00:00:45.000 So, we are going to have to talk about things using multiple dimensions, either in x-y axes or in x-y-z coordinate system, some kind of coordinate system that has more than one dimension, if we going to be talking about the real world. 00:00:45.000 --> 00:00:57.000 All of our vectors are going to be the ideas that we have a change, displacement, when we are moving from one location to another location. 00:00:57.000 --> 00:01:00.000 Velocity, moving at a certain speed. 00:01:00.000 --> 00:01:04.000 But, more than just speed, it is going to talk about the way we are moving. 00:01:04.000 --> 00:01:06.000 Acceleration, when we are changing the velocity. 00:01:06.000 --> 00:01:12.000 Scalars, on the other hand were the things that were just raw length, if we changed it, it is really one-dimensional. 00:01:12.000 --> 00:01:16.000 The distance between two points is, what it would be if you take a tape measure between those two points. 00:01:16.000 --> 00:01:22.000 It does not care what the angle is, it just cares where, how far it is from 'a' to 'b'. 00:01:22.000 --> 00:01:30.000 The displacement on the other hand, would care how did you get there, it is just not the length, there is an entire circle you could go. 00:01:30.000 --> 00:01:35.000 If you were talking about some length, that length could be pointing in any direction on a circle. 00:01:35.000 --> 00:01:40.000 You need more than that if you are going to really talk about displacement. Speed is similar to distance. 00:01:40.000 --> 00:01:49.000 It is just distance/time, if you are looking for the average speed. That is how fast you are traveling, but once again, it is not going to say anything about where you are actually heading. 00:01:49.000 --> 00:01:50.000 Finally, time. 00:01:50.000 --> 00:02:03.000 We are going to treat time as a scalar, however I do want to say that if you are to get in to more heavier Physics, you would wind up seeing that time can actually be treated as another part of the space vector, where we are located. 00:02:03.000 --> 00:02:08.000 That is going to have to do with the idea of space-time, that is beyond what we are talking about right now. 00:02:08.000 --> 00:02:14.000 So, we are going to be able to treat time as just a single dimensional quantity all the time. 00:02:14.000 --> 00:02:16.000 First, before we get started, a quick note on vectors. 00:02:16.000 --> 00:02:20.000 When we are indicating a vector, I like to use a little arrow, like this guy here. 00:02:20.000 --> 00:02:29.000 Other people, they prefer to use a bold font. If you are looking in a text book do not be surprised if you are seeing bold fonts everywhere, I am writing little arrows. 00:02:29.000 --> 00:02:34.000 Or, if you look in one text book, and it has bold fonts, and another has little arrows, now you know why! 00:02:34.000 --> 00:02:36.000 They are both ways of indicating vectors. 00:02:36.000 --> 00:02:41.000 Sometimes, when we already know we are definitely talking about vectors, it is just assumed that we write it like that. 00:02:41.000 --> 00:02:46.000 Finally , when we are writing it with handwriting, I tend to write it like this, because frankly, I am a little bit lazy. 00:02:46.000 --> 00:02:54.000 I could write it like that, but that takes little bit extra effort, so, instead I make this little harpooned hat that lands on top of it. 00:02:54.000 --> 00:03:00.000 So, little harpoon on top makes that little arrow, and that is how I write it when we are dealing with a vector, when we are writing it up. 00:03:00.000 --> 00:03:05.000 But when it is written, it has that actual arrow above it. Alright! So, position. 00:03:05.000 --> 00:03:09.000 Like said before, position is just the location of an object at a given moment in time. 00:03:09.000 --> 00:03:20.000 But instead of having position be just along a single string, it is no longer just a single dimensional coordinate axis, we now would have to have some sort of grid, may be two dimensions, may be three dimensions. 00:03:20.000 --> 00:03:23.000 We are going to have to talk about more than just one dimension though. 00:03:23.000 --> 00:03:30.000 Here, we would be able to talk about the point (3,2), just the x axis distance, and the y axis distance. 00:03:30.000 --> 00:03:36.000 We go over 3, and then we go up 2. Simple as that. 00:03:36.000 --> 00:03:40.000 Distance and displacement. So, like before, distance is a measure of length. 00:03:40.000 --> 00:03:48.000 We got two points 'a' and 'b', the distance is just that line, that straight line distance from one point to the other. 00:03:48.000 --> 00:03:52.000 It is just the length that you would measure if you were using a tape measure, or walking it out with your feet. 00:03:52.000 --> 00:03:58.000 The displacement on the other hand is a vector that indicates the change occurring between those two points. 00:03:58.000 --> 00:04:16.000 So, 'a' to 'b' would be very different vector than 'b' to 'a', and even if we are dealing with the same length over here, but pretend it is the same length, I think it is about the same length, If we had 'c' over here, 'a' to 'c' would be a completely different vector than 'a' to 'b', even though these are the same length. 00:04:16.000 --> 00:04:22.000 So, the length 'ab' = the length 'ac' = the length 'ca' = the length 'ba'. 00:04:22.000 --> 00:04:25.000 But, each one of those would be a totally different vector. 00:04:25.000 --> 00:04:30.000 We are talking about taking a different path, We have to go there in a different way. 00:04:30.000 --> 00:04:36.000 We 6ake the same number of steps, so to speak, to get there, but we are taking a totally different path to get there. 00:04:36.000 --> 00:04:40.000 Because we are going to a different final location. 00:04:40.000 --> 00:04:45.000 For example, say we walk 3 km North of the house, and then 4 km East. 00:04:45.000 --> 00:04:53.000 We start off at home, we go up 3 km, and then we go East another 4 km. 00:04:53.000 --> 00:04:58.000 So, what would our displacement be? Our displacement will be from where we started, to where we ended. 00:04:58.000 --> 00:05:01.000 So, we go like that, and that would be our displacement vector. 00:05:01.000 --> 00:05:11.000 Our displacement vector would be (4,3). 00:05:11.000 --> 00:05:21.000 And also notice, we are not saying this explicitly, but we almost always assume 'up' as positive, and 'right' is positive. 00:05:21.000 --> 00:05:25.000 Because, that is what they are on the x-y axis on the normal Cartesian coordinate system. 00:05:25.000 --> 00:05:29.000 That is what we are used to in Algebra, we tends to translate over. 00:05:29.000 --> 00:05:42.000 Once in a while we might want to change which we consider to be positive, and which we consider to be negative, but for the most part we are going to treat going North as positive, going South as negative, going East, to the right as positive, going West, as negative. 00:05:42.000 --> 00:05:51.000 And if we are dealing with on a flat ground, may be on a table, and had a box, if the box moved to the right, that would be positive, if it moves to the left, that would be negative. 00:05:51.000 --> 00:05:57.000 If the box moved up, that would be positive, if the box moved down, that would be negative. 00:05:57.000 --> 00:05:59.000 It is up to us to impose a coordinate system. 00:05:59.000 --> 00:06:00.000 It is always an important thing to keep in mind. 00:06:00.000 --> 00:06:09.000 It is us humans who impose a coordinate system on the world and make sense where things are, by giving them assigned values. 00:06:09.000 --> 00:06:17.000 The assigned values can vary, but we have to choose how we are going to start with to make the window frame to look at the world with. 00:06:17.000 --> 00:06:22.000 So, our displacement would be this vector right here, (4,3) km. 00:06:22.000 --> 00:06:25.000 We traveled 4 to the East, we traveled 3 North. 00:06:25.000 --> 00:06:27.000 It does not matter which direction we have done it. 00:06:27.000 --> 00:06:34.000 We could have alternatively done it, it does not matter which order we do it in. We could have alternatively done like this, we would have landed at the same spot. 00:06:34.000 --> 00:06:40.000 But what is the distance between where you started and where you ended? We have got a right triangle. 00:06:40.000 --> 00:06:45.000 By Pythagorean theorem, we know that 3^2 + 4^2 has to be equal to, here is the symbol that we use to show distance, when we want to show how long that vector is, we use the magnitude or the absolute value as you are used to seeing in Math. 00:06:45.000 --> 00:07:09.000 So, this would be the absolute value squared, so 9 + 16 = the distance squared, so we get 25, which is going to wind up becoming 5. 00:07:09.000 --> 00:07:19.000 And we would technically get +/- 5, but when we take the square root, we know there is no such thing as a negative value in distance, so we know that we got to have a positive value. 00:07:19.000 --> 00:07:22.000 We can just forget about the negative when we are doing this. 00:07:22.000 --> 00:07:33.000 And finally, what distance did we travel? We noticed that there is a difference between the distance from beginning to end, a path a bird might take, versus the path that we actually took with our feet. 00:07:33.000 --> 00:07:44.000 In this case, if we measure the feet path, we have to go 3 km North, to the first point, and then change to 4 km East. 00:07:44.000 --> 00:07:56.000 So, we would be 3 + 4 =7 for the total km, for the total distance traveled. 00:07:56.000 --> 00:07:58.000 Each one of these are being different things. 00:07:58.000 --> 00:08:05.000 Displacement is the vector that says how do you get from where you started, to where you ended. Which path you have to take. 00:08:05.000 --> 00:08:16.000 You have to, no matter how you do it, no matter what path you wind up taking, you could walk like this, then walk like this, then walk like this, then walk like this, and then walk like this, and get to the point, the same starting point. 00:08:16.000 --> 00:08:19.000 That is going to wind up being up being the displacement vector of zero. 00:08:19.000 --> 00:08:22.000 Because you did not ultimately displace yourself. 00:08:22.000 --> 00:08:30.000 In this case, we did ultimately displace ourselves, we displaced ourselves 4 km to the East, 3 km to the North. Compare that to the distance. 00:08:30.000 --> 00:08:37.000 We wound up being in a different location, we are now away from our original location by 5 km. 00:08:37.000 --> 00:08:42.000 And finally, the feet steps, we had to actually walk using our feet. 00:08:42.000 --> 00:08:51.000 How far we actually traveled by foot, is going to be the total distance we traveled, 3+4 = 7 km. 00:08:51.000 --> 00:08:58.000 Three very different ideas, but important to remember that we can talk about each one of these things, and each one of these is going to get used, at different times. 00:08:58.000 --> 00:09:00.000 Speed and Velocity. 00:09:00.000 --> 00:09:04.000 Just like before, speed is how fast, it is just how fast you are going some time. 00:09:04.000 --> 00:09:09.000 So, it is that length that you have traveled, divided by the time it took you to do that travel. 00:09:09.000 --> 00:09:17.000 But velocity is based on displacement. It asks how you got there, not just how fast you moved to get there, but how you got there. 00:09:17.000 --> 00:09:23.000 Did you go there at this angle, did you go there at this angle, did you go there at this angle. 00:09:23.000 --> 00:09:33.000 Speed, since they are all the same length, if they were all the same length, that would be the exact same speed, but traveling in three very different directions. 00:09:33.000 --> 00:09:35.000 We are going to have different meeting. 00:09:35.000 --> 00:09:44.000 If you are traveling 60 km/h, to the North, that is very different from if you are traveling 60 km/h to the East. 00:09:44.000 --> 00:09:45.000 They would be the same speed, in the pedometer in your car, but they are going to be very different velocities, because you are traveling in a different path, you are traveling in different way, different direction. 00:09:45.000 --> 00:09:58.000 Velocity is based off displacement. Velocity is a vector as well. 00:09:58.000 --> 00:10:07.000 Velocity is the displacement divided by the time, so v = change in displacement/time, Δ d/t. 00:10:07.000 --> 00:10:19.000 Let us go back and look at the example we just had, we started in a house, we traveled 3 km to the North, and then we travel 4 km to the east. 00:10:19.000 --> 00:10:23.000 We do this in 2 hours. 00:10:23.000 --> 00:10:40.000 If it is 2 hours, our displacement, is equal to (4,3) km. 00:10:40.000 --> 00:10:48.000 So, what is our average velocity? Average velocity is the change in our displacement. 00:10:48.000 --> 00:10:52.000 Displacement is the change in the locations, we denote with d. 00:10:52.000 --> 00:11:01.000 We can talk about displacement, as d, like this, but we could also talk as the change in location. 00:11:01.000 --> 00:11:07.000 Once again, we have this thing where location, displacement, sometimes they get used for the same letter, so there is little bit of confusion here. 00:11:07.000 --> 00:11:10.000 But we know we need to talk about how far we traveled. 00:11:10.000 --> 00:11:13.000 We traveled (4,3) km as a vector. 00:11:13.000 --> 00:11:30.000 If we want to find out what the velocity is, velocity = (4,3) km / 2 hours, so velocity is going to be equal to (2,3/2) km. 00:11:30.000 --> 00:11:35.000 Now, if we want to know what our average speed was, we need to see what distance did we travel. 00:11:35.000 --> 00:11:41.000 Distance = 5 km . 00:11:41.000 --> 00:12:02.000 So the average speed is going to be, we are going to look at the magnitude, the size of that speed vector and it is going to be 5/2 = 2.5 km/h . 00:12:02.000 --> 00:12:04.000 Here is an important note. Speed from velocity. 00:12:04.000 --> 00:12:10.000 As we just saw, on the previous example, if you know something's velocity, we can easily figure out its speed. 00:12:10.000 --> 00:12:15.000 You could figure out how far it traveled total and divide it by the amount of time to get there. 00:12:15.000 --> 00:12:21.000 But you can even do better than that. Speed is the length of the velocity vector. 00:12:21.000 --> 00:12:26.000 We are going to figure it out by looking at how long is the velocity vector. 00:12:26.000 --> 00:12:36.000 If we want the speed, we are going to look at the distance that we traveled and divide it by the time, or we could look at the velocity vector, which is displacement/time. 00:12:36.000 --> 00:12:41.000 So, if we just look at the length of the velocity, we already wound up dividing by the time, and it is going to work out. 00:12:41.000 --> 00:12:50.000 Last time, we got that the average speed for that trip was, 2.5 km/h. 00:12:50.000 --> 00:12:59.000 But velocity vector, v = (2,3/2) km/h . 00:12:59.000 --> 00:13:05.000 So, if we want to make it a little bit faster, we just need to see what the magnitude of this is, and it is going to wind up being the exact same as this right here. 00:13:05.000 --> 00:13:09.000 Let us double check that. 00:13:09.000 --> 00:13:38.000 If we do sqrt(2^2 + (3/2)^2), we get sqrt(4 + (9/4)), i.e. sqrt(25/4), which is equal to 5/2, exact same thing. 00:13:38.000 --> 00:13:46.000 So, it is just a question of do we wind up figuring out the distance, and then divide it by the time, or do we wind up figuring out the velocity, which already involves dividing by the time. 00:13:46.000 --> 00:13:50.000 And then just figure out how long that vector is on its own. So, two ways to do it. 00:13:50.000 --> 00:13:56.000 Normally it is going to be easier if we want to find out the speed of something, we know its velocity vector, we just toss it in to this. 00:13:56.000 --> 00:14:10.000 (vx)^2 + (vy)^2, we can easily figure out what our speed is from that. 00:14:10.000 --> 00:14:16.000 Acceleration just comes from velocity. Since velocity is vector, acceleration must be a vector. 00:14:16.000 --> 00:14:20.000 Acceleration = change in velocity / time . 00:14:20.000 --> 00:14:28.000 So, if we have two different velocities in two different times, we find out what the difference between them is, and then we divide it by the time. 00:14:28.000 --> 00:14:32.000 Gravity is going to be just as the same as before, but we remember it is only going to be effective only in one axis. 00:14:32.000 --> 00:14:34.000 There is no lateral gravity on the Earth. 00:14:34.000 --> 00:14:38.000 If we jump in to the air, you do not get shifted to the right or to the left by gravity. 00:14:38.000 --> 00:14:50.000 So, we are not going to have any x-axis shifting and since when we normally talk about coordinate systems, this tends to be down, this tends to be up, and these are right and left. 00:14:50.000 --> 00:14:55.000 Occasionally we will also look from the top-down when we talk about North and South and East and West. 00:14:55.000 --> 00:15:01.000 But if we are talking about something that is lateral and moving vertically, then we are going to normally do it as up and down being the y-axis. 00:15:01.000 --> 00:15:14.000 If up and down is the y-axis, then we need -9.8 m/s/s, because we are going down at 9.8 m/s/s . 00:15:14.000 --> 00:15:19.000 There is not going to any real changes from the formulae before. 00:15:19.000 --> 00:15:24.000 This is the exact same as it was before, except now are winding up looking at it with vectors. 00:15:24.000 --> 00:15:27.000 So, everything is now working in terms of the vectors here. 00:15:27.000 --> 00:15:42.000 Displacement, the location at time t = 1/2 × a t^2 + vit + the initial location. 00:15:42.000 --> 00:15:48.000 This one is a little bit special, in that we are going to wind up having to break it down into its components. 00:15:48.000 --> 00:15:56.000 Remember, before we had, vf^2 = vi^2 + 2a × displacement. 00:15:56.000 --> 00:15:59.000 That was what happened when we were looking in one dimension. 00:15:59.000 --> 00:16:12.000 But if we are now looking in two dimensions, this equation right here applies in each dimension, so we wind up applying it over the x's and then over the y's. 00:16:12.000 --> 00:16:14.000 So we just need to separate it and break it into each one. 00:16:14.000 --> 00:16:17.000 We cannot talk about a vector squared, that does not mean anything. 00:16:17.000 --> 00:16:24.000 We cannot just scale it, because who is going to multiply who? 00:16:24.000 --> 00:16:31.000 Instead we have to break it into its components, then we easily square the components of that vector. 00:16:31.000 --> 00:16:36.000 So, we just keep the x axis, the x components separate from the y components, the y axis. 00:16:36.000 --> 00:16:41.000 They each do their own thing, as long as the acceleration and the distance, they are not changing at all. 00:16:41.000 --> 00:16:46.000 As long as you got a steady acceleration, then we are safe, and we can talk about it this way. 00:16:46.000 --> 00:16:57.000 Acceleration = change in velocity, so now we are just working in terms of vectors, and same as velocity = change in location, or the displacement. 00:16:57.000 --> 00:16:59.000 Lets us look at some examples: 00:16:59.000 --> 00:17:17.000 If we have a car driving North at 30 m/s, and then it turns right, so this is vi, turns right, and the final, goes East at 30 m/s. 00:17:17.000 --> 00:17:24.000 If the car takes 5 s to complete the turn, what is the average acceleration that the car has to have on it for the duration of the turn? 00:17:24.000 --> 00:17:27.000 The first thing to note here, is we have to be talking in multiple dimensions. 00:17:27.000 --> 00:17:34.000 We are talking in multiple dimensions, because the car is moving North and then East, so we doing in two totally different directions. 00:17:34.000 --> 00:17:36.000 We cannot just do this in one coordinate axis. 00:17:36.000 --> 00:17:44.000 So, what we really want to do is, we want to convert these directions and lengths into actual vectors. 00:17:44.000 --> 00:17:58.000 First we are going 30 m/s to the North, so that is going to be positive, so we get, (0,30) m/s is vi. 00:17:58.000 --> 00:18:06.000 vf = 30 m/s to the East. 00:18:06.000 --> 00:18:14.000 If we want to find out what the acceleration is, acceleration = (change in velocity vectors)/(time involved). 00:18:14.000 --> 00:18:58.000 That is, (final - initial)/time = ((30,0) - (0,30)) / 5 = (30,-30) / 5 = (6,-6) m/s/s , because it is changing, for every second it goes, it winds up changing either 6 m/s faster to the East, or -6 m/s 'faster' to the North. 00:18:58.000 --> 00:19:03.000 What that really means is, 6 slower. 00:19:03.000 --> 00:19:15.000 -6, we mean as we are now being accelerated to the South, which means since it is already moving to the North, it is going to lose some of its speed to that southern acceleration. 00:19:15.000 --> 00:19:22.000 Second example is going to be a long one, it is going to create a bunch of different ideas that we are all going to hook together to help us understand how the stuff works. 00:19:22.000 --> 00:19:30.000 We have a ball being thrown out of a window from a height of 10 m, with an initial speed of 20 m/s angled 30 degrees above the horizontal, we can ask some questions. 00:19:30.000 --> 00:19:33.000 What is the initial velocity vector v for the ball? 00:19:33.000 --> 00:19:37.000 Then, ignoring air resistance, how long does it take the ball to hit the ground? 00:19:37.000 --> 00:19:45.000 Finally, ignoring air resistance once again, what is the ball's displacement from its starting point, and what is its distance? 00:19:45.000 --> 00:19:54.000 We have got some building, and a window, somebody throws a ball out of that window. 00:19:54.000 --> 00:20:04.000 We know, if we were to set a horizontal straight line, this is going to be an angle of 30 degrees, and this ball comes out at 20 m/s. 00:20:04.000 --> 00:20:18.000 Down here is the ground, and as time goes on, the ball is going to fly forward, and then gravity is going to take more and more of a share of its velocity, and it is going to eventually land, hit the ground somewhere. 00:20:18.000 --> 00:20:28.000 We want to figure out how long is it going to be flying through the air, how long is it going to take before its y location hits zero. 00:20:28.000 --> 00:20:32.000 Once we know that, we can figure out, how far is it going to make it to the right. 00:20:32.000 --> 00:20:34.000 First, we are going to have to know, what is its initial velocity vector. 00:20:34.000 --> 00:20:47.000 If we have got, something that is 20 long on this side, and 30 degrees here, we can figure out what the other sides have to be, just use trig. 00:20:47.000 --> 00:20:54.000 So, this side, since it is the side opposite, is going to be sin(30) × 20. 00:20:54.000 --> 00:20:59.000 This one, would be cos(30) × 20. 00:20:59.000 --> 00:21:12.000 So, cos(30) × 20 = sqrt(3)/2 × 10 = 17.3 m/s, approximately. 00:21:12.000 --> 00:21:16.000 And sine is going to be 10 m/s. 00:21:16.000 --> 00:21:29.000 That is going to give us initial velocity vector of (17.3,10) m/s. (right, up). 00:21:29.000 --> 00:21:36.000 Now we are ready to move on, because now we have got a vector, and we can make use of this vector, and break each component off, and work with each component separately. 00:21:36.000 --> 00:21:46.000 If we want to figure out how long does it take the ball to hit the ground, we do not care what its x component is, it is the same thing if the ball hits the ground here, or here, or here. 00:21:46.000 --> 00:21:50.000 All that matters is, when does it have that zero. 00:21:50.000 --> 00:21:56.000 When does the ball make contact at the ground, when is the y axis location zero. 00:21:56.000 --> 00:22:13.000 If we have got, d(t) = (1/2)at^2 + vit + initial location, we can then just break this into looking at the y's throughout. 00:22:13.000 --> 00:22:18.000 We just look at the y components throughout, and we can figure out when it is going to hit the ground. 00:22:18.000 --> 00:22:23.000 What is its acceleration? 00:22:23.000 --> 00:22:43.000 The acceleration is, -9.8, so (1/2) × (-9.8) × t^2 + 10t + 10 . 00:22:43.000 --> 00:22:46.000 We want to solve this for when does its location becomes zero. 00:22:46.000 --> 00:22:55.000 We want to make that left side set to zero, and we want to see what 't' will allow that zero to appear on the left side. 00:22:55.000 --> 00:22:59.000 So, -4.9t^2 +10t +10. 00:22:59.000 --> 00:23:03.000 How do we solve something like this, we got a couple of choices. 00:23:03.000 --> 00:23:06.000 Right here, we got a polynomial. 00:23:06.000 --> 00:23:10.000 If we have got a polynomial, one of the first things we can do is factor it. 00:23:10.000 --> 00:23:12.000 To me, that does not look really easy to factor. 00:23:12.000 --> 00:23:20.000 The easiest thing after that, is to chuck it into the quadratic formula, make that machine go through it. 00:23:20.000 --> 00:23:22.000 What is the quadratic formula? 00:23:22.000 --> 00:23:34.000 (-b +/- sqrt(b^2 - 4ac)) / 2a , and that is going to be the solutions when time is going to make it equal to zero. 00:23:34.000 --> 00:23:43.000 That gives us one way to do it, one other way, you could just plug that into a computer or a powerful calculator. 00:23:43.000 --> 00:23:47.000 But let us go through the quadratic formula, because we could all just work through it that way. 00:23:47.000 --> 00:24:15.000 The answers, t = (-10 +/- sqrt(10^2 - 4 × (-4.9) × 10)) / (2× (-4.9)) 00:24:15.000 --> 00:24:50.000 Keep working through that, we get, (-10 +/- sqrt(100 + 196)) / (-9.8) = (-10 +/- sqrt(296))/(-9.8) = (-10 +/- 17.2) / (-9.8) 00:24:50.000 --> 00:24:53.000 Now, we are going to have to ask ourselves, are we going to go with the plus, or are we going to go with the minus. 00:24:53.000 --> 00:25:11.000 Because mathematically, both of those are correct answers, both of them are times when this equation is going to be fulfilled, it will be fulfilled, that equation will be fulfilled both at the plus and the minus. 00:25:11.000 --> 00:25:20.000 But, we can see logically that the ball only hits the ground one side, it does not have both a forward time and a negative time. 00:25:20.000 --> 00:25:28.000 So, what we want to do, is we know our answer can only be a positive time, because this equation is not true at t < 0. 00:25:28.000 --> 00:25:34.000 Before this moment, we do not know where the ball was, the ball was sitting in some apartment before somebody picked it up, and decided to through it out the window. 00:25:34.000 --> 00:25:40.000 At that moment, that equation right here, was not true before time = 0. 00:25:40.000 --> 00:25:46.000 Once time = 0, that is the moment the ball is actually thrown, we know we can start using this equation. 00:25:46.000 --> 00:25:49.000 So, the only solution that will work, is the one where time > 0 . 00:25:49.000 --> 00:26:02.000 We look at this, and we are going to have to find a way for us to have a negative number on top, because (-10 + 17.2) will give us a positive on top, and then we divide by a negative, we wind up getting a negative thing. 00:26:02.000 --> 00:26:11.000 We are going to actually have to go with the negative answer right here, not the negative answer, but the negative of the plus-minus, because it is the only one which would work. 00:26:11.000 --> 00:26:19.000 If we were to solve it out and get both answers, we would be able to get 2 of them, one positive answer and a negative answer. 00:26:19.000 --> 00:26:27.000 But then the negative answer we know, cannot work, so we have to chose which sign is going to give us a time that is positive. 00:26:27.000 --> 00:26:37.000 We chose the negative one, so knock out the positive one, so we are going to get, 2.78 s. 00:26:37.000 --> 00:26:42.000 So that ball has 2.78 s flight time, before it hits the ground. 00:26:42.000 --> 00:26:46.000 With that, we can now figure out what its x location is, when it hits the ground. 00:26:46.000 --> 00:26:54.000 If it has got 2.78 s flight time, we have see how far it has managed to travel in the x, before it hits the ground. 00:26:54.000 --> 00:27:19.000 Its location in the x is just going to be its 17.3 × 2.78 .(velocity × time) 00:27:19.000 --> 00:27:22.000 But, do we have to worry about acceleration? 00:27:22.000 --> 00:27:33.000 The formula that we normally work with, (1/2)at^2, remember, there is no lateral gravity, we do not have anything accelerating the ball once it is thrown, so all we have to worry about is the initial velocity. 00:27:33.000 --> 00:27:53.000 We multiply those out, we get, 48.1 m is the x location, so if we want to combine that, we have got initial location vector is at where, where does it start? 00:27:53.000 --> 00:27:54.000 Does it start at (0,0)? 00:27:54.000 --> 00:27:58.000 No!, remember, the house starts up here, 10 above the ground. 00:27:58.000 --> 00:28:03.000 So we will make the x axis have it be zero at the point of the window it comes out of. 00:28:03.000 --> 00:28:08.000 So zero for the initial location, but it starts at +10. 00:28:08.000 --> 00:28:17.000 Final location, it hits the ground, and it is at 48.1, and then it has got 0 here. 00:28:17.000 --> 00:28:31.000 So the change, the displacement that it experiences between those two locations, is going to be (48.1,0) - (0,10). 00:28:31.000 --> 00:28:42.000 So we are going to get (48.1,-10), because it traveled down 10 m before it hits the ground, is the displacement it experiences. 00:28:42.000 --> 00:28:57.000 If we want to know what the distance it experiences is, remember, we throw it out of the window, and it lands over here, if we want to know what its distance is, we just have to measure, there is the displacement, so we just need to know what is this length. 00:28:57.000 --> 00:29:04.000 So that length is going to be, the size of the displacement vector. 00:29:04.000 --> 00:29:14.000 Size of the displacement vector, sqrt((48.1)^2 + (-10)^2). 00:29:14.000 --> 00:29:21.000 Punch that out, we get, 49.1 m 00:29:21.000 --> 00:29:29.000 So the ball has a displacement vector of 48.1 to the right, and down 10, so -10 on the y axis. 00:29:29.000 --> 00:29:33.000 But the distance between where it lands, and where it started moving is 49.1 m. 00:29:33.000 --> 00:29:41.000 If we want to know how far its travel was, we do not have the mathematical technology yet, we will be able to figure that out. 00:29:41.000 --> 00:29:50.000 We need to go learn some more calculus before we will be able to figure out what its travel path was, how long it had to move through to get where it landed. 00:29:50.000 --> 00:29:55.000 But we can figure what is the distance from where it landed to where it started. 00:29:55.000 --> 00:29:59.000 Hope that all made sense, see you later.