 I called this more projectile problems. This is, I mean, I could argue, you should be able to do all these, these are weird ones, but you're going to find I'm going to start out with the same approach. If it's at an angle components, I'm going to go vertical and horizontal, I'm going to list what I know and then see where that takes me. But first, I want to just jog our memory. What if the projectile is launched horizontally? If the projectile was launched horizontally, what's my initial vertical velocity if we launch horizontally? Zero. One of the most common mistakes with horizontal projectiles is kids put, because there's only one velocity and it's all sideways, it's not a component, they put that in for initial vertical as well. Actually, no, it's easier. It's zero. For a split second, it hasn't started falling yet. All right. Here's a classic question from the provincial exam, as a matter of fact. A projectile is launched from a cliff as shown, check. It lands on the ground 5.2 seconds later. Oh, this time they gave me T. That's going to make some things an awful lot easier, but I bet you that means there's going to be a curve ball later on. A says, find the height of the cliff, B says, find the range, and D says, find the final velocity just before impact. Sean, are we being launched at an angle here? You know what I'm going to do before I do anything else then? I'm going to letter C and rhyme with components. I'm going to find VX and VY initial. There's VX, there's VY initial. I think cosine of 30 is going to be VX over 20, and sine of 30 is going to be VY initial over 20. That's VX and VY initial. I think VY ends up being 10. By the way, I'm not showing my cross-multiply step. I want to test it. It's cross-multiply, right? Although I gave you a dumb way to remember. I think I said something like, VY, sine, that's really, in my homework, I might skip straight to that. Joel, on a quiz or a test, I'll do the trig and do it right just to make sure I don't make a dumb mistake, but I want to get better than I need to be. What do you get for VY initial? It should be, VY should be 10 even, I think, yes? Okay? VX should be 17-point-something, 17.3. Did I ask you to find VX as an answer? In other words, does it say in the question, find VX, say no. Then give it to me to four sig figs. Give me an extra 17-point, what, what? 17.3, what's after the two? Okay, I like that, that's better. Carry a few extra digits because I'm going to be using this, okay? And now, I'm going to do my usual horizontal, vertical, line down the middle. Soon as I write horizontal, I know that horizontal acceleration is zero. Soon as I write vertical, I know that vertical acceleration is negative 9.8. Soon as I also know VY initial is 10, and I also know that VX is 17.32. What else do I know? Ah! This time, they've actually given me T, which is going to be the same in both columns. What was T? I've scrolled down, it was 5-point-something, 5.2, okay. How many things do I know in the horizontal column? Three, I can go to work. How many do I know in the vertical column? Also three, I can go to work, in fact, I can find anything else you want me to. What does part A want me to find? What's the first thing it wants me to find? Height? Is that vertical or horizontal? Okay, I'm looking for an equation that has A, V, I, T, and D in it. Yeah, as a matter of fact, D is by itself already. It's our good old V, I, T plus a half A, T squared. D equals V, I, T plus one half A, T squared, okay. Part A, D, Y equals 10 times 5.2 plus 0.5, negative 9.8, 5.2 squared. Now by the way, this is going to give me the displacement, and since I ended up below from where I started from, I'm kind of expecting a negative answer, I hope. I think I'll do it, do we? Where would you get? Negative 80.5, is there anyone else? Negative 80.5, units, meters. I would accept that technically, if I'm really, really fussy, technically heights a scalar, we don't say how high is the cliff, negative 80.5 meters. I guess technically I should just say that, but if you called it negative, Kale, I'm not taking marks off, I don't mind the vector. Okay, there's A. What does B want us to find? Part B, okay. So over here, all right, DX equals question mark, and I'll ask myself, is there an equation that has A, V, T, and D in it? What's that, Matt? You're saying the same one, but with Xs instead of a horizontal? Except what's the acceleration? Oh, I think what you're really saying, Matt, is this. DX is VXT, which is going to be 17.32 times 5.2, no 1.5 AT squared, because it's horizontal. What's the range? 90. meters, it's part C asking me to find. V final, just before impact. What does V final look like? Well, if I'm imagining the projectile following this path here, I think V final looks something like that. And I'm going to argue that it's made up of a horizontal and a vertical. Is that okay, Connor? By the way, why did I put a final on there, but I didn't bother putting a final on there? Yeah, VX doesn't change. I mean, you can, it's not wrong. Just VX final is VX initial, is VX middle, is VX whenever. So I'm lazy, it's an extra letter. In fact, I can fill in VX. It's 17.32. What about VY final? Now, Emily, if this was ground to ground, VY final would be really easy, because it would be negative 10. Is this ground to ground? Did we launch off a cliff? Or do we just go from the ground to the ground? I think you said a cliff. Okay, so we need to find VY final. So over here, I'm going to think vertical again. And I'm going to say, AY is negative 9.8. V initial is 10. V final equals, I don't know, oh, but I know the time elapsed was 5.2 seconds. Can I find V final from that stuff? Can I find V final from that? Which one? Oh, I got the V by itself already. That's nice, doesn't happen very often. V final equals V initial plus AT. It's going to be 10 plus negative 9.8 times 5.2. What do I get for the final vertical velocity just before impact? 10 plus negative 9.8 times 5.2. Do you guys get negative 40.96? Okay, negative 40.96. So that means over here, I'm going to put 40.96. Why didn't you put the negative, Mr. Dewick? I did, I don't see a negative, Mr. Dewick. There is a negative in that drawing. How does the negative appear in that drawing? It's going down, I've got it in there as a vector though. So thankfully, this is the lovely right angle. How can I find V final, the actual velocity, Pythagoras? And I think as a component, sorry, as a theta, they're going to want this one here, because that's where it's originating from. Let's see. V final equals 17.32. Squared plus 40.96 squared, square root. I get 44.5 meters per second at, well, tangent of the angle is going to be opposite over adjacent. So second function tangent of opposite over adjacent. And I get 67 degrees below the horizontal. 67 degrees below the horizontal. What'd be wrong is with negative 44.5? See, I would argue negative usually means straight down. I probably wouldn't take marks off, but a negative never popped up. The negative is implied with the 67 degrees below the horizontal. Is that a question, Tara, or a stretch? You can make it. Sure, okay. Two more and we're done. Maybe one more, I can't remember. So there's an example of a little twist they can throw at you. Up until now, almost always you've been flying T. Here they told you the time, but the curve ball was, they said part C, which is reason difficult. Find the final velocity. You have to add components together. You have to clue in that it's a three. Vector sum? Number three, a projectile is launched from the ground with an initial velocity of 88 meters per second at an angle of 70 degrees with the horizontal. What's the velocity, magnitude, and direction of the projectile after 4.6 seconds? Do we have an angle here? Sean, you know what I'm gonna do first? It's a word that begins letter C. Okay, let's find VX and DY initial. I'll dull, draw a little picture. 88, where this angle here is 70 degrees. Here's VX, here's DY initial. DY is gonna be sine 88, sine 70. VX is gonna be 88, cos 70. Delusia M, are you okay with that? You good? It's adjacent, hypotenuse and opposite. Do the trick. Trying to cut corners and give you a bit more time to work on the homework. 88, sine 70, 82.69, 88, cos 70, 30.10. Now they want me to find the velocity after 4.6 seconds. Mitchell, and back. Hey, your shots improved. How can I find the velocity? Well, velocity is either gonna look like this, except I gotta be a bit more careful this time, Jacob. It could be on its way down. I don't know, after 4.6 seconds, has it gone halfway through its projectile or not? I don't know. So I'm gonna, first of all, list everything, find parts that I need to, and then I'm gonna draw my vector sum. But right now, I think, don't write this down, I think it's either like that or it's like that. I don't know which one. You know what's gonna tell me when I calculate BY? So I'm thinking this in my mind, but I'm gonna go horizontal, and I'm gonna go vertical. And I'm gonna say the horizontal acceleration is zero. The vertical acceleration is negative 9.8. I'm gonna say that VX is 30.10, and VY initial is 82.69. Oh, and I know one more thing. They gave me the time of flight. Not the time of flight for it to hit the ground, but they said, look, we don't care when it hits the ground, we wanna know what's going on after 4.6 seconds. I think what I'd like to find now is VY final. I'd like to know, vertically, is it on its way up or on its way down? How can I tell, Rhett, if I find the VY final, how can I tell if it's on its way up or on its way down? Positive or negative? Good. Oh, what equation can I use to find VY final? Oh, good gosh, you're saying it's this? Straight plug and chug. It's gonna be 82.69 plus negative 9.8 times 4.6. Once you've written that down, put your pencils down and look up, and I'll show you a shortcut so that you can, at least in your head, ahead of time, know whether you're on your way up or on your way down. The number negative 9.8, what nice number to do math with is that really close to. See, what I would have done is I would have gone negative 10 times the time. You know what negative 10 times 4.6 is? Negative 4.6 is 82 minus 4.6. It's gonna be positive or negative. It's still on its way up. In fact, it's gonna be on its way down sometime around time equals eight seconds because 10 times eight is 80. That's when I might actually end up with a negative answer there. See how I did that? You don't have to, but hey, we're nerds, let's why not, right? What is VY final? Sorry? 37.61 and positive on its way up. Okay, now I can draw my picture. It seems to me the resultant velocity after 4.2 seconds is gonna look like this. And it's going to be made up of a horizontal and a vertical component. And the horizontal component is gonna be 30.10. And the vertical component is going to be, oh, 37.61, yes? Good, on your own right now, find V magnitude and direction. Do the Pythagoras, do the tree. That's what I got. Could be wrong. Opposite over adjacent, shift 10. Gonna take it in right. Opposite over adjacent, shift. I think I'm right. Am I right? No, yes? Yeah. On your test, you can expect some kind of a question or ask you to find a velocity by adding two vectors together, either the impact velocity just forward to the ground or the velocity after a certain number of seconds. It's gonna be above the horizontal if your vector looks like that. It's gonna be below the horizontal if your vector looks like that. And yeah, I guess Vy is the deciding factor. If Vy is positive, it's above the horizontal. If Vy is negative, it's below the horizontal. The last one. Oh. B asks, what's the velocity after 12 seconds? Now I do this, I find Vy, but instead of Vy at 4.6 seconds, you know how much time I put in? 12, I get a negative answer. I end up with a diagram that looked like that. I'm not gonna worry about it. I think you guys are getting the hang of it. Instead, I would like to move to number four. With what final velocity, magnitude and direction? Does the projectile hit the wall? This question, I like this question, I like this question. Emily, why would I say that? Did you hear me say it's gonna be on the test or are you just one of those good students that picks up on little hints that your teacher mentions? I never said it was gonna be on the test. You just picked up on that yourself? Oh, okay. Still three people here. Oh, oh yeah, huh. Someone will explain it to Jacob later. Jacob, what does this question want me to find? In fact, you know what? I can even draw it. I think as a picture, it's gonna look like that just by looking at the diagram they gave me. In fact, this is the same thing that we've just done in the previous question. We're gonna have to find Vx, which is gonna be unchanged. We're gonna have to find a Vy at a certain time, Vy final. Is there an angle here? Let's go components. Woohoo, woohoo, here we go. There's Vx, there's Vy initial. Vx ends up being 26 cos 37. Vy initial ends up being 26 sine 37, 26 cos 37, 26 sine 37, 15.65. So I already know part of this vector triangle up here. If that's my resultant that I hit the wall with, I already know Vx. Vx is 20.76. Katie, you know how I know that's Vx? Because Vx never changes. Right? What about Vy? Is it negative 15.65? No. I'm gonna have to find out how much time has elapsed. Did they give me time in this question? What did they give me in this question? A horizontal displacement. I'm gonna solve for time horizontally. And as soon as I mention that, now I'm gonna do my usual little chart approach. Horizontal, vertical, AX is zero, AY is negative 9.8, Vx is 20.76, Vy initial is 15.65, DX is 40, T equals question mark. What equation has A, V, D and T in it? I think D equals VIT plus a half AT squared. Oh, except woohoo! In fact, I think, and Matt, I think was the one that pointed this out earlier, T is just going to be the displacement divided by the velocity. It's gonna be 40 divided by 20.76. It's a little bit smaller than two. The time that's elapsed is 1.927 seconds. 1.927 seconds. Time that's elapsed is 1.927 seconds. How many different items do I have in the vertical column now? Three I can find whatever they want me to. In this case, we would very much like to, Nicole, find Vy final. By the way, I don't think you've even beyond yet today, well played, or haven't caught it. I'm beyond yawning, that's all right. Okay. Vy final equals, how can I find Vy final? Oh, V final equals V initial plus AT. It's gonna be 15.65 plus negative 9.8 times 1.927. What's the final y velocity? Should be negative, and it should be just, not a big negative, like negative two-ish or something. Negative, negative 3.235. Oh, am I hearing rumbling? Is that right or not? You got negative, and I heard what I heard. I got negative, and I used all the decimals. Two, three, two, I'd take both. If you used all the decimals, that's probably more accurate. I've rounded off here. That's why, by the way, we're still accurate to three sig figs, so that's one of the reasons they go with that. How's that help? Oh, first of all, what's the negative tell me? Going down, so the diagram is correct. This is going to be 3.235, 3.235. You can do it on this triangle. I've rounded a room, I'm gonna quickly redraw the triangle. So here's v final, 20.76, 3.235, v final is gonna be 20.76 squared 20.76 squared plus 3.235 squared equals 21.0, I'll go 21 meters per second. Don't forget your units. At, there's angle theta. How can I figure out angle theta on my calculator? Shift tan, shift tan of what? In this case, opposite adjacent. I think shift tan of 3.235 divided by 20.76. Careful, Mr. Duke. You get 8.9 degrees. Caitlyn, you might be a bit different, but close. At 8.9 degrees below, below the horizontal. There's the first unit. Mostly review, but in two dimensions. There is a little, a couple of comments, or just the one comment here. Feel free to read it. What's your homework? You've got two things to work on. The take home quiz, and remember the very beginning of this unit I handed you, the ultimate vector kinematics review, that'll be due the day of your test. When is the day of your test? A week, 30 days. So you can work on that too. Remember on that ultimate review, if you get stuck, I have answer keys online showing all my steps, as well as the answers are attached to the back. But those are just the correct letters or the correct numbers. The actual steps are online. You good?