 Hey everybody, welcome to Tutor Terrific. This is my fourth video in my unit three of my physics course. We're going to be looking at particular problems and how to do them for two-dimensional kinematics. Projectile motion, of course, is the type of motion you study in your high school physics course or your first level college course if you're going into another field besides physics. We keep it pretty simple here, but the stuff can get pretty dicey when it comes to the math. So we need to practice. So this video, it's all practicing problem-solving skills. Let's get started. I want to start by reminding you of the range formula specifically. We learned that in our last video, but I want to review, here it is, beautiful range formula. I want to review when it's used and when it's not used. So can you use it in the following situation? We are at the top of a cliff and we launch some cannon ball at an angle with respect to the horizontal and it hits somewhere down below. No, you cannot use it here because the initial and final vertical positions are different. This is not level. We call level horizontal range at all. What about this one where we just launch a ball straight horizontal off a cliff onto the ground below? No. Also, not level horizontal range when the position final and initial vertically speaking are over a hundred meters different. So this down here is the situation in which it can be used like this. Okay, there's a bunch of examples. Notice how the starting and ending positions vertically speaking are identical. Even though these launch angles are different for each of these projectiles, their initial and final vertical positions are identical. Even this one that shot straight up. It comes back down to the same exact vertical spot. Okay, now I'm not going to get too much into all of this graph's details. I just want to show you something about the angles specifically. So these projectiles were all launched in this computer simulation at the same initial velocity, but at different angles. Notice that when the launch angle is 90 degrees, there's no horizontal position change at all. As you would expect, this is like our chapter 2 free fall examples. When we start, let's go down here to 15 degrees, a very small angle. We don't get much range. If we increase the launch angle, we get a lot more range. Look at that. From five meters to almost double that to almost nine meters. If you increase again to 45, you get even more range. You get all the way to near 10 meters for this specific example. And notice if we increase beyond 45 to 60 degrees, for example, we start to decrease our range. And notice that 60 degrees and 30 degrees give the same range for the same velocity. That's because complimentary angles, launch angles, give the same range, such as 75 and 15, 30 and 60. And 45 is the winner. It gets you the most range. All right. Now, these problems that I'm going to give you, we're going to do what we normally do, create a set of given and wanted tables. But the first question I ask, besides drawing a picture, because we've already got a picture right over here, is can we use the range formula? Now, in this example, I've got a football being kicked at ground level with a speed of 18 meters per second at an angle of 35 degrees with respect to the horizontal. How far away does it hit? The ground. Okay. First of all, you want to see if the range formula even applies. So we've got a football being kicked on the ground and it lands somewhere else on the ground. So the first question is, can we use the range formula? So because of that truth that the football hits the ground later, when it's kicked from the ground level, you can use the range formula. But would we? Okay? Yes. The reason we would and we can is because we are asked how far away does it hit the ground? That is a range question. It's asking for the difference in horizontal positions of the football when it's kicked and the football when it hits the ground. So this range formula over here is what we are going to use in this problem. Now, when using the range formula, I want you to choose downwards as positive. So the value of g will be positive. The nice thing about the range formula, guys, is that you really don't need to set up your given and wanted table. So we're going to completely skip that step for this problem and just plug in the information given into this formula. Meaning the initial velocity, no components here. This is the total, the angle theta. Notice how if we have to double it in the formula, ng, which would be 9.8 meters per second square when you choose downwards as positive. And your horizontal positive position, your direction, doesn't really matter so much. You would just choose it in the direction that the picture shows the football moving horizontally. Alright, so let's plug this stuff in guys. So we've got v0 equals 18 meters per second, and we've got the angle equals 35 degrees and g, of course, is positive. 9.8 meters per second squared. So here's all that stuff plugged in. You have to square the velocity, the initial velocity, you have to double the angle, and then make sure that g is positive. So you can multiply 18 squared times sine of 70 and then divide by 9.8. And this is what you get. You get 31.1 meters. So that's how far away on the ground the football field that the football hits the ground when kicked with those initial speed and launch angles. Alright, next problem. I've got a diver who runs 18 meters per second. He dives horizontally from the edge of a cliff or diving board or what have you. He reaches the water below three seconds later. Ah, I gave you some time information on this problem. First question I'm going to ask you, sort of part A, which we'll do on this slide, is how high was the cliff? Okay. How high was the cliff? So, is this a situation in which we need a picture? Well, technically, we've got a photo here, but we need a trajectory drawn, which would be drawn in a half parabola fashion from the diving board or what have you into the water below. We do not know the length or the height of the cliff, the length, or which of the distance he was able to get from the edge of this body of water. So, a lot will be left with question mark, but at least we have that trajectory drawn. Now, after that, you ask yourself, can you use the range formula to solve this problem? Well, the problem is asking how high the cliff was. So, definitely not useful for this first part here. But we couldn't use it anyway if we were asked for the distance he was able to get from the edge of the body of water, because it's not level horizontal range. His initial vertical position is much higher than his final vertical position, and so this just doesn't apply, and it's not necessary. Okay, so we need to create our set of given and wanted tables to find the height of the cliff. So, remember, you've got two sets of given tables, one for your horizontal component, one for your vertical component, and you also have two wanted tables, one for your horizontal and vertical components, respectively. So, let's look at this information. 18 meters per second, he runs horizontally. So, that lets you know that the initial horizontal velocity is 18 meters per second. But it's also the final horizontal velocity, because there's no acceleration in the horizontal direction. Also, we can always set the initial horizontal position to zero. Okay. Also, in the y direction, we need to really think about our positive and negative direction before we can continue. So, since he is never traveling upwards, he's always falling, I'm going to choose positive downwards, and based on the picture, he seems to be diving to the left, so I'm going to choose to the left as positive. Okay. Now, back to given y, we know the acceleration due to gravity g is positive, a 9.8 meters per second squared, when downward is positive. We also know that his initial position vertically can be chosen to be zero, and his final, which we don't know, would be positive. We also do know it's hidden in here, that his initial vertical velocity is zero, and that's because he dives out horizontally, so all that 18 meters per second is horizontal. All right. So, here is the table. We also, and besides these things I already mentioned, we also know the time. We know the time for the entire trajectory is three seconds. What we do not know is the final x position, because we don't know how far he lands, away from the edge, and we don't know the height of the cliff. It's just what we're asked to find, y equals, and we don't know the final y velocity, because he's accelerating over some time, but we haven't calculated that yet. But what we really want in this problem is y equals the final position in the vertical sense. So, what is our roadmap? Which equation is going to allow us to find the final y given all of this? Well, take a look at the first equation. It doesn't have y final in it, so we wouldn't use that. The third equation does, but we don't have another variable. We don't have v y final, so we can't use that. So, we have to use the middle one. Notice I didn't even bother with the horizontal equations, because I'm asked for a vertical quantity. So, I'm not going to go over here when I have an equation over here that has one thing unknown, and that's what I want, this middle one. So, two things are zero in here, which is great. We could set y not equal to zero, and we could set v y not t equal to zero, because v y not is zero. So, this is very simple now. It's just the last term. We've seen this before. So, we plug in the time, and we plug in 9.8 meters per second squared. Very simply, we get this. One half times 9.8 meters per second squared times 3.0 seconds squared. Remember to square that time value. And this is what you end up with. When you do that, you get the final y position is 44.1 meters. Anybody catch something wrong here? Look at your initial numbers here. Look at how many digits are significant in each one. There's two in the initial velocity, and there's two in the time. Since we're doing multiplication, we have to have only two sig figs. So, I've got to be careful. Just like everybody, sometimes I make mistakes too. And when I originally made this slide, I made that mistake, and I decided to use that as a learning tool. 44 meters is the real correct answer in the proper number of sig figs for the height of the cliff. Okay. Now the second part of the question. How far away from the base of the cliff did the diver reach the water? Okay. If you understand this, read this carefully, that's asking for x final. How far from the base of the cliff did the diver reach the water? That is the horizontal final position that's requested. Okay. Now we're asked for something like range. Horizontal difference or displacement. Can we use the formula here? It's related to range. We still cannot, because the initial and final vertical positions are different. It's not level horizontal range. So, this does not apply to this problem. All right. So, we need to modify our set of given and wanted tables now that we know the final y position. I'm going to leave all the positive directions the same because it's the same event. It's the same situation. And I'm going to take my y final. Since I know it now, I'm going to delete it from the wanted list and put it in the known given y list. Okay. So, we're asked for x final. Okay. What should we use? Which equation should we use to get x final? Well, look at the second x equation. Do I know the initial x position? Yes. Do I know the initial x velocity? Yes. Do I know the time? Yes. Well, then I can use this to find the final x value over here. So, we're going to plug it zero for x naught. And it's very simple. It's just 18 meters per second times three seconds. That's all this is. And when we plug a chug, we should get 54 meters. All right. So, that's how far he was able to get by running that. That's pretty fast. So, I'm not sure if that's realistic or not. But if it is, that's quite a feat to get 54 meters from the edge of the, wow. I don't even think this guy was going that fast. So, let's see if he was shot by a cannon. I wouldn't be caught dead in that speedo. That's for sure. All right. We already had all that horizontal information. That's what made this so easy, this part B. So, we've done some more problems. I want to review the techniques we've learned so far. So, the first thing I want you to do is determine when you get the problem, when you read it. I want you to draw a picture, determine which objects are involved. That's what happens when you draw the picture. If you've got the person diving off the diving board, make sure you draw the person and the person's trajectory, if they were to do the action that was described in the problem. Secondly, I want you to see if there might be an easy conceptual answer to the problem. There could be in many cases, simple and easy answer that does the trick. Maybe it's not. Usually it's not, it's around our parcel map. Now, before you do your given and wanted table, so I've introduced a new step, and that is to see if the range formula applies to your situation. If the initial and final vertical positions are the same, and you're asked for the horizontal displacement, that would be the range. And if you know the angle and the initial velocity, and you know G, of course, you can use the range formula. So, that would be the end to your situation. If not, then you have to use the other 2D kinematic equations, the set of five. And you'll need to set up your given and wanted tables as we did. Very carefully. And during this time, you just set the positive and negative direction for both your horizontal and vertical motion. So, you get your signs correct, and set up your reference positions where they most help you. Then, you will pick the road map. Which equations can you use, equation or equations, that will most help you get these answers that you need. Always make sure that the equation you use has only one unknown, so you can actually solve it with algebra. Then, you will get to the place where you get the answer requested of you. And that's the set. I'm going to review these next time. This is the set of steps that you need to take. Alright, guys. For that, this video, we're all finished up. I've got one more lesson. We're going to do some more problems, some more complicated problems. But thank you for watching this one. For now, this is Falconator, signing out.