Velocity and Accel in 1 Dimension - Physics 1 Tutor Video

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Velocity and Accel in 1 Dimension - Physics 1 Tutor

Welcome to the Physics Tutor In this DVD course, we are going to discuss the fundamental concepts of Physics and we are going to do it without a bunch of boring lectures and a bunch of derivations that will take you on five boards or whatever to do. What we are going to do it, through working example problems. I have taught a lot of people how to do Algebra, Calculus, Trig, and also Physics. And, in my experience it is actually the toughest subject to actually teach is really Physics and you might say why is that. The reason that I found that people have a tough time with Physics is because most people have a tough time with word problems and that is exactly what this course is, the entire course of Physics is basically just a bunch of Math problems that just happened to be word problems. You pick up a book of Algebra or Trigonometry or something and you will get a problem in it will say, solve this equation and then if you just know how to bang through the different steps, then you can kind of mechanically solve the equation and get the answer as long as you know the rules. Physics, is a little bit different because Physics we are going to give you some rules, but it is not going to be clear how to get to the answer. It is not going to be clear what to do first, what to do second, what to do third, it is not going to be clear how to start the problem. And so, what I am going to teach you in this course, I am going to teach you, in each section, I am going to give you a kind of a basic lecture, a very cut to the chase, do not waste any time overview of the subject that we are talking about. And then, I am going to explain the actual mechanics of how to do the problems by doing problems. And one thing I want to say, is that when you start a Physics problem or when I write a problem up here on the board, and you do not know immediately how to solve it because most of the time you will not, do not freak out, everybody goes through that. You have to learn how to do this stuff by working the problems. And, if you stick with me and start at the beginning and work your way forward, I will promise you that by the end of this DVD course, you will have a good understanding of Physics and you will have a good understanding of how to work problems on Physics. So, without wasting anymore time, let us just get right to it. The first thing that we are going to talk about on this class is the first thing and almost every single Physics book ever printed, and that is One-Dimensional Motion. That is generally going to be the first topic that you have in Physics. Now a lot of times, the very first topic will be a basic chapter that we will talk about, how do you convert units, what are the units in Physics, what is the unit of force, what is the unit of distance, what is the unit of time and stuff like that. I have chosen to kind of skip that, not because it is not important, but I am going to teach you that as we go and we work the problems. So if you do not know how to convert units, do not freak out, we are going to do that as we go along here. What is One-Dimensional Motion anyway? We do not live in a one-dimensional world. We live in a three-dimensional world, as far as the three space dimensions. So in real life, if this pen here is a basketball, let us say, and I move it around up and down, I can move it left and right, that is one dimension, well let us call it the X dimension. I can also move it to you and from you, and that is another dimension of space and we could call that the Y dimension, let us say. Well, we can also have freedom of movement in this up and down direction here and we can call that the Z dimension. They are just labels, it does not really matter; X, Y and Z, that is what we call the three-dimensional motion, that is the real world. When you go to the grocery store or actually a better example is when you fly in an airplane, you might turn left, you might turn right, but you might also go up and down, so that is three-dimensional motion. When you start to learn Physics, three-dimensional motion is too complicated, so you need to start out with one-dimensional motion. What does that mean? That means, imagine that there were a string across the string. Here is just a regular piece of string and there were some beads on that string, you know that can had a hole on them and they could slide back and forth. One-dimensional motion just means motion constrained to move in only one dimension, because we are going to simplify the problems and we are going to look the simplified version of the world. So, instead of talking about X and Y and Z, because that is pretty complicated and we are going to get there. Let us do not do that at first, let us start just by talking about one of those motions, one of those dimensions, and it does not matter which one we usually just talked about the X dimension. This entire section is going to talk about how the things move when you describe the motion in one dimension. So, we are going to talk about the X dimension a lot and just remember that is just a label, it just means motion constrained in one direction only; backwards or forwards, no up no down, no to and from, just along this one line. The first thing that we need to talk about is the concept of displacement. All right, now I ask you, what do you think that word means, displacement? Displacement means to displace something, to move something. That is exactly what it means. Physics is just going to define this formal definition of what displacement is. Let us say for the sake of argument, you had a ruler over here. This is a ruler and this is zero, one, two, three, four, five, six, and something like that. This is just a measurement like the inches; this could be the inches along the ruler. And, let us say you have some, some fly or something like that, it was sitting right here on top of the number three, and then over some period of time, the fly or the bug or whatever it is just kind of walks over to the right and eventually it ends up over here at the number six. So here, I am going to put an I here, this is where the bug starts at the initial position that is why I put an I here, and over here, I am just going to put an F because this is the final position of the bug and it kind of walks over here. Well, I should ask you a pretty simple question, how far did the bug move or in terms of Physics, what was the bug’s displacement? Well, how far did it displace? It is pretty simple. It started at three; it ended up at six, so I think you will all agree that the bug moved three units to the right. We say Delta X is equal to six minus three, which is just simply equal to three. Now, I need to stop for a second and explain what this means. I told you this is—we are going to be talking about motion in one dimension which most of the time, we just use the label of X to describe motion in one dimension, motion along the X direction. Now, what this Delta thing mean? This is a Greek symbol Delta and basically, it just means change. When you see a Delta there, it means the change in something. So, I could have the change and the position. Along the X direction, I could have the change of the velocity of a car, I can have the change in the acceleration of a spaceship, I could have a change in a lot of different things. But anyway, this Delta here, this triangle, it just means change, so anytime you see a triangle, all you have to do is put the word change in front of it and then you will know exactly what that is trying to tell you. We know from everyday life that Delta X, which is the displacement; is just three. So, I am going to define displacement a little bit more formulating in terms of Physics and in terms of what you will see in a textbook; Delta X is equal to X final minus X initial. Not too bad right? I mean, this is exactly what we have here. I am just generalizing it for you. I did not want to throw this up on the board initially because some people are put off by formulas. I am trying to show you from an example. So, in order to find that how far something moved, all you do is you take the final position of that thing and subtract the initial position and you have got the displacement, you know how far it has moved. The next thing we wanted to talk about is the velocity. You all know what the term velocity means. It means basically, how fast is something moving, the velocity of a car, the velocity of a spaceship, very similar to the concept of speed. And in fact, they are almost exactly the same, the only difference is the term velocity; means that not only do you know how fast that is going, it means, you know what direction it is going too. Are you going five miles an hour this way or you are going five miles an hour this way? So, speed would just be telling you how fast you are going and velocity is how fast are you going and in what direction you are going. So, most of the time, we are going to talk about velocity because we are going to be trying to figure out in our problems, are we going this way or are we going this way and how fast are we going. So, that is basically what velocity is. I think we all have a fairly good understanding and big picture of what the term velocity actually means. So the backup for one second, the unit of displacement is meters. As I said, we are going to talk about units as we go through this stuff in Physics here and so I am teaching as we go. The unit of distance in Physics that we typically use in books today is the unit of meters. It is about three feet, a meter is about a yard. So, just deal in terms of meters. So, the next thing we are going to talk about is velocity, which we already said is basically speed. So, what is the unit of velocity? Well, in a car, you know that the unit is miles per hour where miles is a unit of distance and hour is a unit of time. In Physics, you are almost always going to be dealing in meters per second,. And by the way, I am putting this stuff in brackets because that is just my notation. I like to put the units in bracket because it kind of reminds me that they are separate, they are different and just put them off to the side, put them in brackets. In that way, you do not get mixed up and think this is some variable or something. It is not something that you have to do. It is something I do. So, the unit of velocity is meters per second in Physics. The next thing we are going to talk about is the concept of average velocity. What if you had, I am going to draw a plot here, this is time, the time axis, and this is the X-axis. So, remember, we are talking motion along one dimension. So, as time goes on, I am plotting the position of this little bead or whatever it is in this direction like that. And, let us just say that at the beginning of time, T is equal to zero. When I start looking at the whole environment here, I start out like this and the plot looks something like that and I am going to explain this in a second. What this means is that, when I started looking at my stopwatch, when started looking at my bead that I am looking at and my motion on this one dimension, I started moving initially. Because, if you look as a function of time, I started to kind of move along X pretty quickly, but then as time went on, I kind of flat down and I slowed down, because you see over here, as time goes on, I am not moving very far along X. As you go along time in this region over here, I am not moving very far along X, so I have kind of slowed down whereas initially I kind of—I was speeding up pretty fast because as time goes on here, I have actually covered quite a bit of distance along X. And so, in the points of between here, I am kind of slowing down. So, basically, what I am doing here is I am slowing down. So, this motion will look something like this, if you would look at it. When I started off, I am going pretty fast and then I am slowing down like this, so let us look at it again. I am going pretty fast and then I am slowing down. It is just plotting the position as the function of the time. Now, at each point along this graph, I have a different velocity. Here I am going pretty fast relatively speaking because as time goes on, I am covering a lot of distance along X and over here that we have already talked about, I have kind of slowed down. Now in the middle, obviously, my velocity is changing. So, the concept of the average velocity comes in here and that is what I am trying to tell you here. You can have motion that has a different velocity, you could be slowing down, you could be speeding up. So, you can have different velocities all along your motion here, but you can also define what we call the average velocity which is exactly what it sounds like. It is like, you have this big picture motion going on and you are changing your speed, but we can define in Physics what we call the average velocity, which is kind of, if you look at the whole motion what the average velocity is there along the motion. We will define that in terms of the velocity with the bar over it which just means average velocity and that is going to be equal to how far did I moved, totally, from start to finish and how long did it take me to move here. So, if I want to expand to the top, Delta X, remember I said Delta just means final minus initial, so that is the final position minus the initial position and Delta T just means the change in the time which is the final time minus the initial time. This is the definition for average velocity. And if you also look at this, because we have defined it this way, the units of average velocity is also meters per second because we have the unit of distance from the top and the unit of time on the bottom and of course, this fraction will stay there, so meters per second makes sense with the units. What this means is that if you have a motion where your velocity is changing, all you do is you look at where you started, which in this case is the initial position here, where you ended up, which is the final position here and how long did it take you to accomplish the motion. And then, you just do this division and you can end up with the average velocity and that basically means on the whole, if you had to pick one constant speed to represent this changing speed here, then that would be the average. Sometimes you are going faster, sometimes you are going slower, but in the average, you can define it average velocity like this and it only depends on the initial and the final values and how long it takes to get there. As an example, I mean, just as a really quick and dirty simple example, which I will just kind of write here. Let us say you traveled some distance, along the X-axis of 10 feet and we almost never deal with feet in Physics, but in this case, I am just going to do it just for the help of it, and how long does it take you to go 10 feet. It takes you two seconds to go 10 feet. So, then you can define an average velocity which is just going to be Delta X over Delta T. And, that is just a fancy way of saying; how far did you moved in the X direction, I have already given you that; that is 10 and how long did it take you to do that, it take you two seconds. You do this division and you get an answer of five, but you always got to be clear; five what? Well, I was dividing feet by seconds, so it is going to be feet per second, good deal. Now, we are going to work a little bit different kind of problem and we are going to get some practice with converting the units. We are going to get some practice with converting units and we are going to get some practice with some of this initial-final velocity stuff. You have a problem at the bottom of your screen now. And basically, the problem says that for a marathon, there is a record out there that says it takes someone two hours, nine minutes, and 21 seconds to do this marathon and the distance is 26 miles, 385 yards. So, what we need to determine the speed which is basically like the velocity in miles per hour. That is what we want to solve for. And what we are given is that the time is two hours, nine minutes, 21 seconds. I am going to make this clear here, 21 seconds and the distance is 26 miles, 385 yards, and we are trying to find the answer here in miles per hour. And, this is basically your problem in unit conversions here and then we are also going to use the formula for speed. So, we know that speed or basically velocity, like I said, they are almost exactly the same concepts,, is going to be distance over time. That is another way of saying Delta X over Delta T. This is just the change in your distance, the change in the time, so we are given this information, we know the distance, we know the time, so what is the big problem here? Well obviously, the problem is we are given the distance in 26 miles and then 385 yards, so this is much smaller than a mile, and the time is given in hours, minutes, and seconds, and we want to compute things in hours. So, what we really need to do in order to solve this problem, in order to get the final units we want is to convert this distance into miles. So, there is no yards anymore, it is just miles and then also to convert this time to hours, and then we will have miles and then we will have hours and we can simply divide them, we will get miles per hour right. So, let us go ahead and do that. For the distance, we already have 26 miles, so we are going to hang onto that for a second. What we want to do is we want to convert this 385 yards to miles. So, 385 yards, that is what we are starting with and what we want to convert is this yards to miles. Now, I am going to show you a trick. This is probably one of the most useful things that I am going to teach you in the whole class on how to convert units is this right here, it is not really a trick, it is just the actual way to which you convert units. You are starting with yards and you want to get to miles. What I am going to show you how to do is the following. I am going to write it on the board, you may not quite understand it, but then I am going to explain it as we go. What I know is that one yard is equal to three feet, that is something that you should know or that you probably can look up. What I also know is that one mile is equal to 5280 feet and this is going to allow me to convert yards to miles, how? Because, and again, you may not quite understand this yet, I am going to explain it to you here in just a second. These yards cancel with these yards, this feet cancels with this feet. So in the end, if I were to take the 385 times the three divided by, because it is on the bottom; divided by the 5280, then, the only things left is the miles, because the yards went out with the yards, the feet went out with the feet. What I have done here is I have set up a set of conversion factors. What you do is you write down what you start with, you draw your little horizontal line and then you start picking conversion factors in order to cancel what you do not want to have in the end, which is yards; you want to get rid of yards. I knew I had to go from yards to miles but I did not really know up to the top of my head how many yards where in the miles so I kind of did it in two steps. Yards to feet, feet to miles and arranged them—you know, if I would have to put this on the top; the 5280 on the top, feet and the miles on the bottom, then I would not have cancelled the feet. Because if I would have feet up here and feet over here and I would not be able to cancel. In order to cancel, you have to have one on the top and one on the bottom. And, we are going to do this over and over again so you are going to get the hang of it. Essentially, when you do this, 385 times three divided by 5280, is going to give you a grand total of 0.21875 miles. And this is how many miles, 385 yards is equal to. So, the total distance is equal to the 26 miles I had to start with, plus this 0.21875. So in the end, the distance is 26.219 miles and I am kind of rounding a little bit here. So, what we have done is we have taken the 26 miles, 385 yards, we have converted that to 26.219 miles. The 385 yards gives you 0.219 miles. Now, for the time, and we are going to do the same thing here and I think you will get the hang of it. I want to end up with hours, I have got hours, minutes and seconds. So, what I need to do is convert minutes and seconds into hours and then I can have the entire thing in hours. So, let us start with the seconds first, 21 seconds and I want it to convert to hours. Now, I know that one minute is equal to 60 seconds, so I am going to put one minute up on the top, 60 seconds on the bottom. Why did I put seconds on the bottom? Because, what I want to do is I want to cancel the seconds, so I want to take this and cancel it with that. But, I do not want to stop in minutes because what the whole point of this was must trying to get over to hours, so what I am going to do is I am going to say, there is also 60 minutes in one hour. Why did I put the minutes in the bottom and the hours on the top? Because I wanted to cancel minutes with minutes, I will start it with seconds and everything else is cancelled and I have just got hours left over. So, if I take 21 divided by 60, divided by another 60; so 21 divided by 60, I would get 0.0058 and that is hours. That is how many hours of 21 seconds is equal to. Now, let us work on the minutes; nine minutes, 60 minutes in one hour and that is really all I have to do because minutes cancelled with minutes, so I am converting nine minutes to hours. So, nine times one divided by 60 is 0.15 hour. So, let us not lose track the big picture. The time is two hours, nine minutes, 21 seconds, 21 seconds is equal to these mini hours; nine minutes is equal to these mini hours. So,, the total time is equal to two hours, that I started with, plus the nine minutes which is equal to 0.15, plus the 21 seconds, 21 seconds is equal to this 0.0058, so the total time is 2.1558 hours, that is the total. Now, I did all of these and these, why, because it is an exercise on how to convert units. I want to calculate the speed in miles per hour, so I need to have some distance in miles and now I have that, which is right here. This is how many miles I traveled and I need to have time in hours which is these mini hours right here, all right. So, the answer, which I am going to kind of work on kind of right here, the velocity is equal to distance over time, which is equal to 26.219 divided by time which is 2.1558 and this is in miles and this is in hours, so then the final deal here is going to be 12.2 miles per hour. That was a lot of junk there on the board. We had time, we had distance, we just have to convert various things into miles and into hours. When we have the miles and we have the hours, we could divide the two and we get miles per hour. And I think you can see that by working these kinds of problems that Physics is going to be kind of like that. I mean in general, it is really going to be a bunch of tedious stuff. And really now, it was a fairly simple problem, fairly mechanical, we knew basically what to do. And, as you go on, sometimes it is not going to be clear what to do, so the challenge is going to be figuring out what to do. The next thing I want to talk about is the concept of average of acceleration and also average acceleration. You know a minute ago we have talked about position and displacement and we have talked about velocity and we talked about average velocity. Now, we are going to talk about acceleration. Acceleration is the phenomenal that you have whenever you are speeding and you are slowing down. So, if you are kind of moving along in a constant speed here, you do not have any acceleration, your acceleration is zero. If you are speeding up or you are slowing down, you are accelerating, just like in your car. So, then you can also define an average acceleration, just like we could define an average velocity and so we will write that as acceleration with a bar over it. And, that one is equal to the change of velocity over the change in time, the same as what we had talked about before. So, this will be the final velocity minus the initial velocity over the final time minus the initial time. And of course velocity, the units of velocity we have already talked about is in meters per second and for the units, the unit of time on the bottom is seconds, so what you can write this is as meters per second times one over second. Meters per second which is this quantity here times one over seconds because the S is on the bottom, so what you end up with this meters per second squared, and that is why you see in your book the units of acceleration as meters per second squared exactly because of this right here. So, if I were going to kind of plot what this is really talking about, then you would have some time and then you would have some velocity and then if you were going to start looking here, you would kind of have the velocity would be going up and then going down, and then going up. So basically here, you are accelerating and here you are turning back around and going the other way, and here you are accelerating again and you are going back the other way. So obviously, your velocity is changing here, you are speeding up, you are slowing down, et cetera. But, if you are going to find some average velocity, it would depend only upon the endpoints here because that only depends on the initial and the final values. And, what you are doing here is you have some complicated motion that—where you are speeding up and you are slowing down. But, on the whole, on the average, you might be speeding up, you might be slowing down, but there is an average acceleration to it, so you can define it average quantity that will describe big picture here and that is only going to depend on the initial and the final value which is exactly what we have right there. So, let us do another problem. In this problem, we are going to send a man to the man and we are going to do it in a really weird way. We have a cannon basically and we are going to shoot this guy out of the cannon and you see its final velocity is 10.97 kilometers per second and the cannon is 220 meters long. And so, what we want to do is really want to find out how fast is that guy accelerating out of the cannon here. Now, I like also just mention that in your Physics book, you will probably see a lot of these stuff here we are dealing in terms of kilometers, we have been talking about meters this far. A meter is about that big, just about off the screen here, it is about three feet or something. A kilometer is a kilometer, which is 1000 meters, so it is pretty far a little distance here. I am just trying to give you a scale for what we are talking about here. First thing we need to do, when you are confronted with any Physics problem, the number one thing you need to do is draw a picture. Most people, if you just draw the picture, the solutions will kind of pop out, so what we do is we have a cannon here, this is the wheel I guess should say of the cannon, I am not a really big artist here, and but you have some cannon, right like this. And, this cannon is a pretty big cannon, it is 220 meters long and then you have got some guy, that I was kind of put out here like this get pops out of the cannon, he shot out of the cannon, and he shot at a final velocity of 10.97 kilometers per second. So, he shot out of the cannon, he is flying at 10.97 kilometers per second. That is pretty fast when you think about it, 10 kilometers every second, I mean, that is like really, really fast. Obviously, when he was inside of the cannon over here, his initial velocity was just zero. He was not moving when he started. When he was in the cannon initially, he was not moving at all, and then the cannon fires, he is then accelerated down this guy right here, he is accelerating faster and faster and finally, he pops out and goes off to the moon. What we want to do is we want to figure out what is the acceleration and the other way to say that is what is the average acceleration that this guy experienced. In other words, here he started to speed up and he popped out of the cannon, what we want to do is we want to figure out what is the average acceleration. Now, I have already given you the basic big picture formula for that, and that is Delta V, Delta T. So, all we had to do is we had to figure out how fast—well, I mean just blow it up here. This is the final velocity minus the initial velocity, final time minus the initial time. So, if we know the initial and final velocities then we know how long it took him to get to the end, then we can calculate his average acceleration, so let us see if we can do that. His final velocity is the velocity that he had when he popped out the other side of the cannon so it is 10.97. The initial velocity we already said it was zero. Now, how long did it take him to actually travel down to the end of the cannon, what is his final time and his initial time, in other words, how long did it take him to actually do this. Well obviously, there is nothing in the problem that tells you that and so, you might be stuck. This is a great in Physics when you start scratching your head and you are like I do not know what to do now because they did not give me enough information. And then, you start saying well maybe, there is a problem with the problem, maybe the book made a mistake, maybe they did not give me what I need to know. Do not fall for that stuff. There is always enough information. Now here, I have reserved a time for you to figure out what you need to figure out. I deliberately worked the problem like this to kind of go in logical order to show you how you can reach a point when you do not know what to do next. What we need to do now is to start looking at the problem a little more carefully and figure out how can we figure out the time it took this guy to go from one end to the next. Take note that I am going to draw a little line here, we are going to come back to this point and find this time right now. Notice that we know the final velocity of this guy is 10.97, the initial velocity is zero. So, we know he starts at zero and he accelerates to a final velocity. So, if you assume that his acceleration was uniform, in other words, he kind of uniformly accelerated up this tube, then you might be able to say that the average velocity here how fast on average was he traveling down the tube is just simply going to be the average of these two numbers. So, Vf plus Vi over two. That is literally the definition of an average. You take two numbers and you unite them together and you divide it by two. So, 10.97 plus zero divided by two is—and, when you put this in your calculator, you will get 5.485 kilometers per second. So, initial velocity, final velocity, average velocity is on the average, how fast was he traveling, sometimes he was going faster, sometimes he was going slower but this is the average. So, if this is the average speed and this is the length of the tube, then we can use these two numbers to calculate how long it takes him to go to the end. Now remember, velocity is equal to distance over time. So time, by rearranging this equation is equal to distance over velocity and you can see that because this is an equation like any other equation, I can multiply both sides by T. And so, I will have VT is equal to D, if I multiplied both sides by Trump University and then, to solve for time, I can just divide it by V, so I get T is equal to D over D. So, I just solved this equation for time, that is all I did there, I am just trying to calculate the time here and I know the distance the guy traveled, I know his average velocity, I can calculate the time. So, the time is equal to distance over velocity. So, what we want to do is we want to solve for the time, but what we want to do is the distance, the length of this tube is 220 meters, but I do not want to put 220 in this equation. Because, if I put 220 meters in this equation and I divide it by this velocity, this velocity is in kilometers per second, and so, what I am trying to do is I am trying to arrive at the unit of seconds. I am trying to get the time and I have in compatible units here because I have got the tube in meters and I have got the velocity in kilometers per second and that is not going to wash. The first thing you have to notice, the first rule of Physics is the units have to be consistent. So, I need to either convert this to meters per second or I need to convert this to kilometers and it is easy for me to convert this to kilometers, so I am just going to put a decimal point here 0.220 and this is kilometers and then the velocity is 5.485 and this is kilometers per second. Now, the kilometers will cancel, the seconds will flip back onto the top and in the end, the time that takes this guy that pop out the tube is 0.04 seconds. One thing I am going to explain to you real fast, 220 meters, you can convert just like anything else, you have to use a conversion factor, so 1000 meters is one kilometer, same as everything else. You take the 220 and divide it by 1000 and that gives you 0.220 kilometers. You know I kind of did that in my head there, but I want to explain to you where it comes from. So, do not forget the big picture. We knew the initial and final velocity is here, we needed to calculate the time. In order to calculate the time, we calculated an average speed for this guy flying down the tube which is just the average of the initial and final velocities. And, we knew the distance of the tube, so knowing the formula of velocity is distance over time, then we basically plugged in everything then we solved for time, comes out to 0.04 seconds. So, we put that in like this and then the acceleration. So, on the top, we have 10.97 minus zero and this is in kilometers, the bottom is in terms of seconds, so the average acceleration of this guy down the tube is 10.97 divided by 0.04 which is 274.25 kilometers per second. So, you put a guy in the cannon, you shoot him out the end and he is going of acceleration of 274.25 kilometers per second squared. The units is kilometers per second squared because I had the velocity on the top, kilometers per second, this is the time on the bottom, so I have kilometers per second, per second which is this guy. Now, if you wanted to compare this to something that is familiar to you, the acceleration of gravity, the acceleration due to gravity is called G. The acceleration that holds us to the ground is 9.8 meters per second squared. Of course, this is meters per second squared, this is kilometers per second squared. So, how would you compare those two numbers, 274.25 kilometers per second squared, I am going to convert this to meters per second squared, 1 kilometer is 1000 meters. So, kilometers cancels with kilometers, so this acceleration basically just multiplied by a thousand, so you have 274,250 meters per second squared. Now, view the same two numbers, this is just in terms of kilometers per second squared and this is in meter per second squared. I just converted that to show you that look at how incredibly huge this number is compared to the acceleration of the gravity; the gravity that is holding me right now on the floor talking to you today. So, shooting yourself out of a cannon is a pretty dangerous thing and it would kill anybody obviously doing that in terms of this problem right here.

Math Tutor DVD

Math Help, Algebra Help, Calculus Help, Physics Help and everything in between.

http://www.mathtutordvd.com/

Welcome to the Physics Tutor

In this DVD course, we are going to discuss the fundamental concepts of Physics and we are going to do it without a bunch of boring lectures and a bunch of derivations that will take you on five boards or whatever to do. What we are going to do it, through working example problems. I have taught a lot of people how to do Algebra, Calculus, Trig, and also Physics. And, in my experience it is actually the toughest subject to actually teach is really Physics and you might say why is that. The reason that I found that people have a tough time with Physics is because most people have a tough time with word problems and that is exactly what this course is, the entire course of Physics is basically just a bunch of Math problems that just happened to be word problems. You pick up a book of Algebra or Trigonometry or something and you will get a problem in it will say, solve this equation and then if you just know how to bang through the different steps, then you can kind of mechanically solve the equation and get the answer as long as you know the rules. Physics, is a little bit different because Physics we are going to give you some rules, but it is not going to be clear how to get to the answer. It is not going to be clear what to do first, what to do second, what to do third, it is not going to be clear how to start the problem.

And so, what I am going to teach you in this course, I am going to teach you, in each section, I am going to give you a kind of a basic lecture, a very cut to the chase, do not waste any time overview of the subject that we are talking about. And then, I am going to explain the actual mechanics of how to do the problems by doing problems. And one thing I want to say, is that when you start a Physics problem or when I write a problem up here on the board, and you do not know immediately how to solve it because most of the time you will not, do not freak out, everybody goes through that. You have to learn how to do this stuff by working the problems. And, if you stick with me and start at the beginning and work your way forward, I will promise you that by the end of this DVD course, you will have a good understanding of Physics and you will have a good understanding of how to work problems on Physics. So, without wasting anymore time, let us just get right to it.

The first thing that we are going to talk about on this class is the first thing and almost every single Physics book ever printed, and that is One-Dimensional Motion. That is generally going to be the first topic that you have in Physics. Now a lot of times, the very first topic will be a basic chapter that we will talk about, how do you convert units, what are the units in Physics, what is the unit of force, what is the unit of distance, what is the unit of time and stuff like that. I have chosen to kind of skip that, not because it is not important, but I am going to teach you that as we go and we work the problems. So if you do not know how to convert units, do not freak out, we are going to do that as we go along here.

What is One-Dimensional Motion anyway? We do not live in a one-dimensional world. We live in a three-dimensional world, as far as the three space dimensions. So in real life, if this pen here is a basketball, let us say, and I move it around up and down, I can move it left and right, that is one dimension, well let us call it the X dimension. I can also move it to you and from you, and that is another dimension of space and we could call that the Y dimension, let us say. Well, we can also have freedom of movement in this up and down direction here and we can call that the Z dimension. They are just labels, it does not really matter; X, Y and Z, that is what we call the three-dimensional motion, that is the real world. When you go to the grocery store or actually a better example is when you fly in an airplane, you might turn left, you might turn right, but you might also go up and down, so that is three-dimensional motion.

When you start to learn Physics, three-dimensional motion is too complicated, so you need to start out with one-dimensional motion. What does that mean? That means, imagine that there were a string across the string. Here is just a regular piece of string and there were some beads on that string, you know that can had a hole on them and they could slide back and forth. One-dimensional motion just means motion constrained to move in only one dimension, because we are going to simplify the problems and we are going to look the simplified version of the world. So, instead of talking about X and Y and Z, because that is pretty complicated and we are going to get there. Let us do not do that at first, let us start just by talking about one of those motions, one of those dimensions, and it does not matter which one we usually just talked about the X dimension.

This entire section is going to talk about how the things move when you describe the motion in one dimension. So, we are going to talk about the X dimension a lot and just remember that is just a label, it just means motion constrained in one direction only; backwards or forwards, no up no down, no to and from, just along this one line.

The first thing that we need to talk about is the concept of displacement. All right, now I ask you, what do you think that word means, displacement? Displacement means to displace something, to move something. That is exactly what it means. Physics is just going to define this formal definition of what displacement is. Let us say for the sake of argument, you had a ruler over here. This is a ruler and this is zero, one, two, three, four, five, six, and something like that. This is just a measurement like the inches; this could be the inches along the ruler. And, let us say you have some, some fly or something like that, it was sitting right here on top of the number three, and then over some period of time, the fly or the bug or whatever it is just kind of walks over to the right and eventually it ends up over here at the number six. So here, I am going to put an I here, this is where the bug starts at the initial position that is why I put an I here, and over here, I am just going to put an F because this is the final position of the bug and it kind of walks over here. Well, I should ask you a pretty simple question, how far did the bug move or in terms of Physics, what was the bug’s displacement? Well, how far did it displace? It is pretty simple. It started at three; it ended up at six, so I think you will all agree that the bug moved three units to the right.

We say Delta X is equal to six minus three, which is just simply equal to three. Now, I need to stop for a second and explain what this means. I told you this is—we are going to be talking about motion in one dimension which most of the time, we just use the label of X to describe motion in one dimension, motion along the X direction. Now, what this Delta thing mean? This is a Greek symbol Delta and basically, it just means change. When you see a Delta there, it means the change in something. So, I could have the change and the position. Along the X direction, I could have the change of the velocity of a car, I can have the change in the acceleration of a spaceship, I could have a change in a lot of different things. But anyway, this Delta here, this triangle, it just means change, so anytime you see a triangle, all you have to do is put the word change in front of it and then you will know exactly what that is trying to tell you.

We know from everyday life that Delta X, which is the displacement; is just three. So, I am going to define displacement a little bit more formulating in terms of Physics and in terms of what you will see in a textbook; Delta X is equal to X final minus X initial. Not too bad right? I mean, this is exactly what we have here. I am just generalizing it for you. I did not want to throw this up on the board initially because some people are put off by formulas. I am trying to show you from an example. So, in order to find that how far something moved, all you do is you take the final position of that thing and subtract the initial position and you have got the displacement, you know how far it has moved.

The next thing we wanted to talk about is the velocity. You all know what the term velocity means. It means basically, how fast is something moving, the velocity of a car, the velocity of a spaceship, very similar to the concept of speed. And in fact, they are almost exactly the same, the only difference is the term velocity; means that not only do you know how fast that is going, it means, you know what direction it is going too. Are you going five miles an hour this way or you are going five miles an hour this way? So, speed would just be telling you how fast you are going and velocity is how fast are you going and in what direction you are going. So, most of the time, we are going to talk about velocity because we are going to be trying to figure out in our problems, are we going this way or are we going this way and how fast are we going. So, that is basically what velocity is. I think we all have a fairly good understanding and big picture of what the term velocity actually means.

So the backup for one second, the unit of displacement is meters. As I said, we are going to talk about units as we go through this stuff in Physics here and so I am teaching as we go. The unit of distance in Physics that we typically use in books today is the unit of meters. It is about three feet, a meter is about a yard. So, just deal in terms of meters. So, the next thing we are going to talk about is velocity, which we already said is basically speed.

So, what is the unit of velocity? Well, in a car, you know that the unit is miles per hour where miles is a unit of distance and hour is a unit of time.
In Physics, you are almost always going to be dealing in meters per second,. And by the way, I am putting this stuff in brackets because that is just my notation. I like to put the units in bracket because it kind of reminds me that they are separate, they are different and just put them off to the side, put them in brackets. In that way, you do not get mixed up and think this is some variable or something. It is not something that you have to do. It is something I do. So, the unit of velocity is meters per second in Physics.

The next thing we are going to talk about is the concept of average velocity. What if you had, I am going to draw a plot here, this is time, the time axis, and this is the X-axis. So, remember, we are talking motion along one dimension. So, as time goes on, I am plotting the position of this little bead or whatever it is in this direction like that. And, let us just say that at the beginning of time, T is equal to zero. When I start looking at the whole environment here, I start out like this and the plot looks something like that and I am going to explain this in a second. What this means is that, when I started looking at my stopwatch, when started looking at my bead that I am looking at and my motion on this one dimension, I started moving initially. Because, if you look as a function of time, I started to kind of move along X pretty quickly, but then as time went on, I kind of flat down and I slowed down, because you see over here, as time goes on, I am not moving very far along X. As you go along time in this region over here, I am not moving very far along X, so I have kind of slowed down whereas initially I kind of—I was speeding up pretty fast because as time goes on here, I have actually covered quite a bit of distance along X. And so, in the points of between here, I am kind of slowing down. So, basically, what I am doing here is I am slowing down. So, this motion will look something like this, if you would look at it. When I started off, I am going pretty fast and then I am slowing down like this, so let us look at it again. I am going pretty fast and then I am slowing down. It is just plotting the position as the function of the time.

Now, at each point along this graph, I have a different velocity. Here I am going pretty fast relatively speaking because as time goes on, I am covering a lot of distance along X and over here that we have already talked about, I have kind of slowed down. Now in the middle, obviously, my velocity is changing. So, the concept of the average velocity comes in here and that is what I am trying to tell you here. You can have motion that has a different velocity, you could be slowing down, you could be speeding up. So, you can have different velocities all along your motion here, but you can also define what we call the average velocity which is exactly what it sounds like. It is like, you have this big picture motion going on and you are changing your speed, but we can define in Physics what we call the average velocity, which is kind of, if you look at the whole motion what the average velocity is there along the motion.

We will define that in terms of the velocity with the bar over it which just means average velocity and that is going to be equal to how far did I moved, totally, from start to finish and how long did it take me to move here. So, if I want to expand to the top, Delta X, remember I said Delta just means final minus initial, so that is the final position minus the initial position and Delta T just means the change in the time which is the final time minus the initial time. This is the definition for average velocity. And if you also look at this, because we have defined it this way, the units of average velocity is also meters per second because we have the unit of distance from the top and the unit of time on the bottom and of course, this fraction will stay there, so meters per second makes sense with the units. What this means is that if you have a motion where your velocity is changing, all you do is you look at where you started, which in this case is the initial position here, where you ended up, which is the final position here and how long did it take you to accomplish the motion. And then, you just do this division and you can end up with the average velocity and that basically means on the whole, if you had to pick one constant speed to represent this changing speed here, then that would be the average. Sometimes you are going faster, sometimes you are going slower, but in the average, you can define it average velocity like this and it only depends on the initial and the final values and how long it takes to get there.

As an example, I mean, just as a really quick and dirty simple example, which I will just kind of write here. Let us say you traveled some distance, along the X-axis of 10 feet and we almost never deal with feet in Physics, but in this case, I am just going to do it just for the help of it, and how long does it take you to go 10 feet. It takes you two seconds to go 10 feet. So, then you can define an average velocity which is just going to be Delta X over Delta T. And, that is just a fancy way of saying; how far did you moved in the X direction, I have already given you that; that is 10 and how long did it take you to do that, it take you two seconds. You do this division and you get an answer of five, but you always got to be clear; five what? Well, I was dividing feet by seconds, so it is going to be feet per second, good deal.

Now, we are going to work a little bit different kind of problem and we are going to get some practice with converting the units. We are going to get some practice with converting units and we are going to get some practice with some of this initial-final velocity stuff. You have a problem at the bottom of your screen now. And basically, the problem says that for a marathon, there is a record out there that says it takes someone two hours, nine minutes, and 21 seconds to do this marathon and the distance is 26 miles, 385 yards. So, what we need to determine the speed which is basically like the velocity in miles per hour. That is what we want to solve for. And what we are given is that the time is two hours, nine minutes, 21 seconds. I am going to make this clear here, 21 seconds and the distance is 26 miles, 385 yards, and we are trying to find the answer here in miles per hour. And, this is basically your problem in unit conversions here and then we are also going to use the formula for speed. So, we know that speed or basically velocity, like I said, they are almost exactly the same concepts,, is going to be distance over time. That is another way of saying Delta X over Delta T. This is just the change in your distance, the change in the time, so we are given this information, we know the distance, we know the time, so what is the big problem here?

Well obviously, the problem is we are given the distance in 26 miles and then 385 yards, so this is much smaller than a mile, and the time is given in hours, minutes, and seconds, and we want to compute things in hours. So, what we really need to do in order to solve this problem, in order to get the final units we want is to convert this distance into miles. So, there is no yards anymore, it is just miles and then also to convert this time to hours, and then we will have miles and then we will have hours and we can simply divide them, we will get miles per hour right. So, let us go ahead and do that.

For the distance, we already have 26 miles, so we are going to hang onto that for a second. What we want to do is we want to convert this 385 yards to miles. So, 385 yards, that is what we are starting with and what we want to convert is this yards to miles. Now, I am going to show you a trick. This is probably one of the most useful things that I am going to teach you in the whole class on how to convert units is this right here, it is not really a trick, it is just the actual way to which you convert units. You are starting with yards and you want to get to miles. What I am going to show you how to do is the following. I am going to write it on the board, you may not quite understand it, but then I am going to explain it as we go. What I know is that one yard is equal to three feet, that is something that you should know or that you probably can look up. What I also know is that one mile is equal to 5280 feet and this is going to allow me to convert yards to miles, how? Because, and again, you may not quite understand this yet, I am going to explain it to you here in just a second. These yards cancel with these yards, this feet cancels with this feet.

So in the end, if I were to take the 385 times the three divided by, because it is on the bottom; divided by the 5280, then, the only things left is the miles, because the yards went out with the yards, the feet went out with the feet. What I have done here is I have set up a set of conversion factors. What you do is you write down what you start with, you draw your little horizontal line and then you start picking conversion factors in order to cancel what you do not want to have in the end, which is yards; you want to get rid of yards. I knew I had to go from yards to miles but I did not really know up to the top of my head how many yards where in the miles so I kind of did it in two steps. Yards to feet, feet to miles and arranged them—you know, if I would have to put this on the top; the 5280 on the top, feet and the miles on the bottom, then I would not have cancelled the feet. Because if I would have feet up here and feet over here and I would not be able to cancel. In order to cancel, you have to have one on the top and one on the bottom. And, we are going to do this over and over again so you are going to get the hang of it.

Essentially, when you do this, 385 times three divided by 5280, is going to give you a grand total of 0.21875 miles. And this is how many miles, 385 yards is equal to. So, the total distance is equal to the 26 miles I had to start with, plus this 0.21875. So in the end, the distance is 26.219 miles and I am kind of rounding a little bit here. So, what we have done is we have taken the 26 miles, 385 yards, we have converted that to 26.219 miles. The 385 yards gives you 0.219 miles.

Now, for the time, and we are going to do the same thing here and I think you will get the hang of it. I want to end up with hours, I have got hours, minutes and seconds. So, what I need to do is convert minutes and seconds into hours and then I can have the entire thing in hours. So, let us start with the seconds first, 21 seconds and I want it to convert to hours. Now, I know that one minute is equal to 60 seconds, so I am going to put one minute up on the top, 60 seconds on the bottom. Why did I put seconds on the bottom? Because, what I want to do is I want to cancel the seconds, so I want to take this and cancel it with that. But, I do not want to stop in minutes because what the whole point of this was must trying to get over to hours, so what I am going to do is I am going to say, there is also 60 minutes in one hour. Why did I put the minutes in the bottom and the hours on the top? Because I wanted to cancel minutes with minutes, I will start it with seconds and everything else is cancelled and I have just got hours left over. So, if I take 21 divided by 60, divided by another 60; so 21 divided by 60, I would get 0.0058 and that is hours. That is how many hours of 21 seconds is equal to.

Now, let us work on the minutes; nine minutes, 60 minutes in one hour and that is really all I have to do because minutes cancelled with minutes, so I am converting nine minutes to hours. So, nine times one divided by 60 is 0.15 hour. So, let us not lose track the big picture. The time is two hours, nine minutes, 21 seconds, 21 seconds is equal to these mini hours; nine minutes is equal to these mini hours. So,, the total time is equal to two hours, that I started with, plus the nine minutes which is equal to 0.15, plus the 21 seconds, 21 seconds is equal to this 0.0058, so the total time is 2.1558 hours, that is the total.

Now, I did all of these and these, why, because it is an exercise on how to convert units. I want to calculate the speed in miles per hour, so I need to have some distance in miles and now I have that, which is right here. This is how many miles I traveled and I need to have time in hours which is these mini hours right here, all right. So, the answer, which I am going to kind of work on kind of right here, the velocity is equal to distance over time, which is equal to 26.219 divided by time which is 2.1558 and this is in miles and this is in hours, so then the final deal here is going to be 12.2 miles per hour. That was a lot of junk there on the board. We had time, we had distance, we just have to convert various things into miles and into hours. When we have the miles and we have the hours, we could divide the two and we get miles per hour. And I think you can see that by working these kinds of problems that Physics is going to be kind of like that. I mean in general, it is really going to be a bunch of tedious stuff. And really now, it was a fairly simple problem, fairly mechanical, we knew basically what to do. And, as you go on, sometimes it is not going to be clear what to do, so the challenge is going to be figuring out what to do.

The next thing I want to talk about is the concept of average of acceleration and also average acceleration. You know a minute ago we have talked about position and displacement and we have talked about velocity and we talked about average velocity. Now, we are going to talk about acceleration. Acceleration is the phenomenal that you have whenever you are speeding and you are slowing down. So, if you are kind of moving along in a constant speed here, you do not have any acceleration, your acceleration is zero. If you are speeding up or you are slowing down, you are accelerating, just like in your car. So, then you can also define an average acceleration, just like we could define an average velocity and so we will write that as acceleration with a bar over it. And, that one is equal to the change of velocity over the change in time, the same as what we had talked about before. So, this will be the final velocity minus the initial velocity over the final time minus the initial time. And of course velocity, the units of velocity we have already talked about is in meters per second and for the units, the unit of time on the bottom is seconds, so what you can write this is as meters per second times one over second. Meters per second which is this quantity here times one over seconds because the S is on the bottom, so what you end up with this meters per second squared, and that is why you see in your book the units of acceleration as meters per second squared exactly because of this right here.

So, if I were going to kind of plot what this is really talking about, then you would have some time and then you would have some velocity and then if you were going to start looking here, you would kind of have the velocity would be going up and then going down, and then going up. So basically here, you are accelerating and here you are turning back around and going the other way, and here you are accelerating again and you are going back the other way. So obviously, your velocity is changing here, you are speeding up, you are slowing down, et cetera. But, if you are going to find some average velocity, it would depend only upon the endpoints here because that only depends on the initial and the final values. And, what you are doing here is you have some complicated motion that—where you are speeding up and you are slowing down. But, on the whole, on the average, you might be speeding up, you might be slowing down, but there is an average acceleration to it, so you can define it average quantity that will describe big picture here and that is only going to depend on the initial and the final value which is exactly what we have right there.

So, let us do another problem. In this problem, we are going to send a man to the man and we are going to do it in a really weird way. We have a cannon basically and we are going to shoot this guy out of the cannon and you see its final velocity is 10.97 kilometers per second and the cannon is 220 meters long. And so, what we want to do is really want to find out how fast is that guy accelerating out of the cannon here. Now, I like also just mention that in your Physics book, you will probably see a lot of these stuff here we are dealing in terms of kilometers, we have been talking about meters this far. A meter is about that big, just about off the screen here, it is about three feet or something. A kilometer is a kilometer, which is 1000 meters, so it is pretty far a little distance here. I am just trying to give you a scale for what we are talking about here.

First thing we need to do, when you are confronted with any Physics problem, the number one thing you need to do is draw a picture. Most people, if you just draw the picture, the solutions will kind of pop out, so what we do is we have a cannon here, this is the wheel I guess should say of the cannon, I am not a really big artist here, and but you have some cannon, right like this. And, this cannon is a pretty big cannon, it is 220 meters long and then you have got some guy, that I was kind of put out here like this get pops out of the cannon, he shot out of the cannon, and he shot at a final velocity of 10.97 kilometers per second. So, he shot out of the cannon, he is flying at 10.97 kilometers per second. That is pretty fast when you think about it, 10 kilometers every second, I mean, that is like really, really fast. Obviously, when he was inside of the cannon over here, his initial velocity was just zero. He was not moving when he started. When he was in the cannon initially, he was not moving at all, and then the cannon fires, he is then accelerated down this guy right here, he is accelerating faster and faster and finally, he pops out and goes off to the moon.

What we want to do is we want to figure out what is the acceleration and the other way to say that is what is the average acceleration that this guy experienced. In other words, here he started to speed up and he popped out of the cannon, what we want to do is we want to figure out what is the average acceleration. Now, I have already given you the basic big picture formula for that, and that is Delta V, Delta T. So, all we had to do is we had to figure out how fast—well, I mean just blow it up here. This is the final velocity minus the initial velocity, final time minus the initial time. So, if we know the initial and final velocities then we know how long it took him to get to the end, then we can calculate his average acceleration, so let us see if we can do that. His final velocity is the velocity that he had when he popped out the other side of the cannon so it is 10.97. The initial velocity we already said it was zero. Now, how long did it take him to actually travel down to the end of the cannon, what is his final time and his initial time, in other words, how long did it take him to actually do this.

Well obviously, there is nothing in the problem that tells you that and so, you might be stuck. This is a great in Physics when you start scratching your head and you are like I do not know what to do now because they did not give me enough information. And then, you start saying well maybe, there is a problem with the problem, maybe the book made a mistake, maybe they did not give me what I need to know. Do not fall for that stuff. There is always enough information. Now here, I have reserved a time for you to figure out what you need to figure out. I deliberately worked the problem like this to kind of go in logical order to show you how you can reach a point when you do not know what to do next.

What we need to do now is to start looking at the problem a little more carefully and figure out how can we figure out the time it took this guy to go from one end to the next. Take note that I am going to draw a little line here, we are going to come back to this point and find this time right now. Notice that we know the final velocity of this guy is 10.97, the initial velocity is zero. So, we know he starts at zero and he accelerates to a final velocity. So, if you assume that his acceleration was uniform, in other words, he kind of uniformly accelerated up this tube, then you might be able to say that the average velocity here how fast on average was he traveling down the tube is just simply going to be the average of these two numbers. So, Vf plus Vi over two. That is literally the definition of an average. You take two numbers and you unite them together and you divide it by two. So, 10.97 plus zero divided by two is—and, when you put this in your calculator, you will get 5.485 kilometers per second.

So, initial velocity, final velocity, average velocity is on the average, how fast was he traveling, sometimes he was going faster, sometimes he was going slower but this is the average. So, if this is the average speed and this is the length of the tube, then we can use these two numbers to calculate how long it takes him to go to the end. Now remember, velocity is equal to distance over time. So time, by rearranging this equation is equal to distance over velocity and you can see that because this is an equation like any other equation, I can multiply both sides by T. And so, I will have VT is equal to D, if I multiplied both sides by Trump University and then, to solve for time, I can just divide it by V, so I get T is equal to D over D. So, I just solved this equation for time, that is all I did there, I am just trying to calculate the time here and I know the distance the guy traveled, I know his average velocity, I can calculate the time.

So, the time is equal to distance over velocity. So, what we want to do is we want to solve for the time, but what we want to do is the distance, the length of this tube is 220 meters, but I do not want to put 220 in this equation. Because, if I put 220 meters in this equation and I divide it by this velocity, this velocity is in kilometers per second, and so, what I am trying to do is I am trying to arrive at the unit of seconds. I am trying to get the time and I have in compatible units here because I have got the tube in meters and I have got the velocity in kilometers per second and that is not going to wash. The first thing you have to notice, the first rule of Physics is the units have to be consistent. So, I need to either convert this to meters per second or I need to convert this to kilometers and it is easy for me to convert this to kilometers, so I am just going to put a decimal point here 0.220 and this is kilometers and then the velocity is 5.485 and this is kilometers per second. Now, the kilometers will cancel, the seconds will flip back onto the top and in the end, the time that takes this guy that pop out the tube is 0.04 seconds. One thing I am going to explain to you real fast, 220 meters, you can convert just like anything else, you have to use a conversion factor, so 1000 meters is one kilometer, same as everything else. You take the 220 and divide it by 1000 and that gives you 0.220 kilometers.

You know I kind of did that in my head there, but I want to explain to you where it comes from. So, do not forget the big picture. We knew the initial and final velocity is here, we needed to calculate the time. In order to calculate the time, we calculated an average speed for this guy flying down the tube which is just the average of the initial and final velocities. And, we knew the distance of the tube, so knowing the formula of velocity is distance over time, then we basically plugged in everything then we solved for time, comes out to 0.04 seconds. So, we put that in like this and then the acceleration. So, on the top, we have 10.97 minus zero and this is in kilometers, the bottom is in terms of seconds, so the average acceleration of this guy down the tube is 10.97 divided by 0.04 which is 274.25 kilometers per second.

So, you put a guy in the cannon, you shoot him out the end and he is going of acceleration of 274.25 kilometers per second squared. The units is kilometers per second squared because I had the velocity on the top, kilometers per second, this is the time on the bottom, so I have kilometers per second, per second which is this guy. Now, if you wanted to compare this to something that is familiar to you, the acceleration of gravity, the acceleration due to gravity is called G. The acceleration that holds us to the ground is 9.8 meters per second squared. Of course, this is meters per second squared, this is kilometers per second squared. So, how would you compare those two numbers, 274.25 kilometers per second squared, I am going to convert this to meters per second squared, 1 kilometer is 1000 meters. So, kilometers cancels with kilometers, so this acceleration basically just multiplied by a thousand, so you have 274,250 meters per second squared.

Now, view the same two numbers, this is just in terms of kilometers per second squared and this is in meter per second squared. I just converted that to show you that look at how incredibly huge this number is compared to the acceleration of the gravity; the gravity that is holding me right now on the floor talking to you today. So, shooting yourself out of a cannon is a pretty dangerous thing and it would kill anybody obviously doing that in terms of this problem right here.

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