 So what we're looking at here is Projectile launched off of a cliff. This is the most difficult type of a question and there are a couple of strategies The first strategy is to find the time of flight to the top because you know what the top v y is zero Then find how long it takes to get all the way back down to the ground and Then add the two times together Yeah Strategy two is to do it all at once. That's my preferred method. Yes, it involves a bit of math, but it's shorter Strategy three I don't recommend but some kids like it Alex. It's find the time to get to the top Double it that'll give you the time to get to the bottom back level with the cliff and then figure out using your whatever Your v y is right here. How fast you would fall to the very very bottom at all three times together? Mr. Do it prefers Strategy number two The goal for almost all of these Alexis is going to be find the time of flight once in a while They'll give you the time in which case the question falls apart because most of our equations have a T in them But if they don't give us the time of flight almost always our first thing is going to be I'd find T How break things up into vertical and horizontal like example 10 so example 10 says find the flight time We're launching off of a 60 meter high cliff in my wonderful diagram at an angle of 30 degrees with a velocity of 50 meters per second Cassidy, are we off of a cliff? Say yes. Are we at an angle? Say yes. So I cannot use that 50 Because it's neither horizontal nor vertical first thing I have to do is I have to break this up into its components Now for me, I find this diagram a little bit small. So I'm gonna redraw it It's really up to you, but I'm big on big diagrams. I'm gonna do this over here There's well, I'm gonna try and do with a straighter line than that. Mr. Do it Let's try that again. Hello back to here There's the 50 and I'm gonna break that up into Horizontal and vertical once again, why did I put an initial here but not here? Yeah, yeah Vx isn't changing why because ax is Zero a y is changing So I'm always want to reinforce that because person the most common mistake I see kids make is they want to put that negative 9.8 Which is vertical into the horizontal because they don't like that. There's no acceleration like please. I must hurry Continue I'm gonna break this up into its components. Oh and the angle here is 30 degrees Right there So I'm gonna quickly label my triangle Spencer this Vx opposite adjacent or hypotenuse. I Think I agree, but I'd say with more authority and certainty this Vy opposite or a hypotenuse Opposite That's the hypotenuse What do you want to find first Spencer Vy or Vx doesn't matter but Which trig function are we going to use to find Vy? Oh And H which trig function was that? You know if you had a good teacher He would have given you some kind of a stupid way last class perhaps maybe to remember that Vy Was a specific trig function Vy was So Vy was sign and there's probably some dumb way that you can remember that if you had a good teacher He might have told you that last class, but apparently you didn't have a good teacher last class So yeah, I know the feeling believe me. Yeah as it turns out Vy well it ends up being this the sign of 30 ends up being Vy over 50 Sherman if I do a little cross-multiply thingy. I end up with this Vy equals 50 Sign 30. I think the sign of 30 is point five. I think this is actually exactly 25 so I'm gonna check it yet 30 degrees. I have memorized because it comes up so often Spencer redeem yourself which trig function am I gonna use to find Vx? Which trig function as the a and the h in it? Cosine fact, you know what it's gonna be exactly this but instead of sign, you know, it's gonna appear there So I'm just gonna not bother writing the fraction I'm gonna be clever just in my homework will take shortcuts like this right and I'm gonna go straight to clueing in hey It's coast Cosine of 30 is 0.866 for what it's worth, but I don't know what that 1050 is 40 something Give it to me to for sig fig because this is not my final answer and I'm gonna be using this to find other stuff Of what if you okay, so 43.3 0 43.3 was actually the 466 fair enough. I will never use this 50 Now that I've used it to find Vy and Vx that 50 is dead to me as far as I'm concerned In fact, what I'm gonna do is I'm now gonna draw a little line down the middle of my page I'm gonna call one side Horizontal and I'm gonna call one side Vertical and as soon as I do that Tyson There's a second thing that I do that to me is almost a no-brainer like I want it to become a habit. I go AX equals And a vertical acceleration and horizontal acceleration and I know both of these Justin, what's the horizontal acceleration kiddo? Too slow, what's the horizontal? It needs to be almost that quick horizontally. What's happening? What did we say? What's the horizontal acceleration? Zero I get to fill that in. What's the vertical acceleration Justin always make it a 9.8. You said Okay, I also emphasize that because now hey my questions no longer blank. I feel better got wrote something written down Oh, and there's two more things I can fill in I can write Vx and I can write Vy initial turns out Vx and I'm just reading this as 43.3 and Vy initial is 25 and Right now. I can't go any further because I have two things in each column and you can't solve anything in physics with only two pieces of information I need something else so I'm gonna go back to the diagram I'm gonna look very very carefully and I'm gonna say what did they tell me that I haven't used yet What did they tell me that I haven't used yet? just D y what? Yeah, what is it as a number? I need to disagree with first of all it is a vertical I know I'm looking at the picture sometimes They'll give you DX the range which actually makes the question way way way way way way way way easier because your quadratic cancels But if they give you d y and it's a clip and it's an angle We're gonna be stuck with a quadratic equation and d y Jessica redeem yourself is not 60. What is it really? Why negative 60 I? Ended up below from where I started now. I would say hey Which column do I know three things in? I'm gonna solve this vertical and I'm gonna solve this by finding T time of flight How well I'm gonna find an equation that has an a of v a d and a t in it And I'm actually gonna cheat because I know I'm gonna need more room. So I'm gonna go zoom. Oh, isn't that cool? What equation has a a v a d and a t in it? Oh specifically a vi which I guarantee one of them does Yo The old standard the quadratic one the squared t squared d Equals vi T plus one half a t squared and Chris I'm just gonna emphasize The fact this is vertical to remind me to make sure I put the vertical stuff in there It's funny. I didn't do that last block and I accidentally put the horizontal velocity right there because my brain But I trip on my brain And I'm gonna plug in all my numbers So I know that dy is negative 60 25 t minus 4.9 T squared What? Where'd the negative 4.9 come from I took one extra step in my head Yeah Half a half of nine point eight is four point half of negative nine point eight is negative four point nine That's one worth memorizing as a shortcut. Otherwise, you got to rewrite the whole silly equation just to do that step And let's let's do that math in our head What kind of an equation is this? Quadratic Alex, how do I know? It's a squared. How do I solve the quadratic? What's the first thing I have to do? Before I do anything else Make it equal to zero now. I have a personal preference I hate it when my squared term is negative. So even though plusing this to this side will be the easiest I'm actually gonna plus this over and minus this over. I'm gonna rewrite this as positive 4.9 t squared plus 25 t Minus 60 Equals zero it'd be great if this factor. No, it's got decimals. It's not gonna factor in fact Justin That's gonna be a that's gonna be B and that's gonna be C Now put your pencils down who has a graphing calculator in this class You may have a quadratic solver if you haven't use it. Where is the quadratic solver you ask? Two buttons to the start in the top left corner two buttons over three buttons down you have an app button. Yeah Yes Shouldn't it be what negative 25 who gets the candy back there? It's so does Like I said, I trip on my brain all the time it's all Back to my question Who has a graphing calculator? So if you press the app button Polysmolt pol y smlt you might have polysmolt version two I don't know which works fairly similarly But if you run it and I think you have to press any key and then I think it says option one is polynomial root finder and Then somewhere along the way it asks you for the degree or the power up to and Then see if you can figure it out from there and go ahead now if you're going to use that on a quiz or a test I'll give you zero unless you write down the quadratic formula right underneath this then I have to assume that you use the quadratic formula I'll give you some marks because I probably do a provincial and I'm okay with the technology Some of you may have a scientific calculator that has a quadratic solver If you do great use it with the same provision that I just told now write down the quadratic equation some of you do not if You're good with the quadratic equation It's the fastest way to get the answer If you're a little shaky on the quadratic equation I will in just a second show you an alternate method That does not require you to use the quadratic formula and it gets you there in two extra lines Okay, but for now I Would do something like this somewhere off in the margin For me maybe right here I would write x equals negative b plus or minus the square root of b squared minus 4 a c All over two way. Do I have any German exchange students in this class? No, okay, because they in Europe they use a different thing with p's and q's and a different method So I always have to explain that In fact, I think t is going to be Positive 25 because it's negative b and b is already negative plus or minus the square root of negative 25 squared minus 4 times a times c all over To a and I know that 2 times 4.9 is 9.8, which is a number that's fairly familiar to me You use your solver. What do you get? By the way ignore the negative answer, right? That's wrong. I think it's wrong So as I said ignore the negative answer Sesame Street is brought today by the word ignore as In most of the time I will choose to ignore you Okay What's the positive answer, please? Can you give it to me to three sig figs because we're gonna use we don't want to round this off We're gonna use it to find other stuff, too. I would definitely go 6.88 Now I deliberately haven't typed this in up here because I find if I type it in too many of you Just copy what I'm typing and don't know how to type this into your calculator yourselves So if you did this and you didn't get 6.88 hit me up later And I'll show you how to type it in with by the way You need a bracket there and you need a bracket there inside the square root and a bracket there another bracket there but I'll let you figure that out yourselves or You can do this without using the quadratic formula. Oh, by the way Now as soon as I did that I would go running over here and I would say oh Horizontally the time is also 6.88 seconds because the time it takes for me to hit the ground vertically That's how far how long I'm moving horizontally for Right. How can you do this without using the quadratic formula simple? Put your pencils down and look up Unfortunately, I don't have enough room to show this step because I need room here to find the time to the top And I'm running out of space. I'm gonna try and do this a little bit small I'm gonna go like this or Instead of finding t which is what I wrote down here right away You could find Vy final How fast going vertically at impact now? Can I find it? Let's see. Is there an equation that has a the initial D and V final in it Is there an equation that has a the initial D and V final in it Is there some run now you're done. You any which one the squared one the squared one Which conveniently actually has Vy final by itself already Kind of nice. Oh, well it has Vy final squared by itself already. So if you want to you can go like this when I type that in the initial was 25 squared plus two times negative 9.8 times negative 60 Negative what try that again the six didn't go get it negative 60 I Get 18 oh one square root and And I get Vy final is forty two point four four and I'm wrong Why am I technically wrong? Why is Vy final not forty two point four four? Alex Okay, I'm gonna have to remember that it's going down It's negative. In fact those of you that are in math 12 You've already heard me rant about when you square root both sides plus or minus This is one time where you actually want the minus not the plus Why does that help Andrew? Well now that I have V final I Know the initial I know a I Can find T. Is there an equation that has V final V initial a and T in it Which one? Get the T by itself Which is one we've been doing almost since grade 10 if you go T equals Vf minus Vi over a If you go Bracket negative forty two point four four minus twenty five. There's my V final my V initial divided by negative nine point eight There it is six point eight eight like nap got Did I use the quadratic equation? Nope, and it was probably only a line or two longer. I Don't care which method you use I use the quadratic because I've got solvers built into any calculator that I purchase And I'm lazy that way on my answer keys what you'll often see if you're looking online is all right Use qf which stands for quadratic formula, but you won't see any work Which means I cheated and use my solver because I was doing it at home late at night And I just didn't feel like showing every step Anyways There are two ways to get that six point eight eight with the quadratic formula or find VY final negative and Then solve for two Meanwhile We did what they asked We found T What is part B want us to find Spencer What was the last word you said the word top would that imply vertical or horizontal here? Do you think I think we're gonna tackle this vertically as well? We have our vertical data listed here a VI and D. I Know one more thing. Oh, I don't know D actually D is where I ended up At the top. I don't know how high I am right now. I know Ay is negative nine point eight and VY initial is 25 but I do know one more thing at The top what else do I know? just Vy final is zero does that mean you're stopped at the top? No means you're moving horizontally still Eric But for a split second vertically you've come to a stop And I want to find time Do I have an equation that has a VI VF and T in it? Well, yes, I do in fact You know what it's the same one that Andrew read to us about 30 seconds ago. It's sitting right here in red It's gonna be VF minus VI over a The time to the top is gonna be zero minus 25 All over negative nine point eight. I'm sorry Christian for writing so small What is zero minus 25 divided by negative nine point eight? How many of you want to do good in this course? All of you raise your hands. I hope Can I give you some advice as someone who's taught this course now for quite a while? The most common mistakes by far that I see on tests tests are stupid calculator mistakes kids have written all the correct numbers And they have the wrong answer Almost inevitably those are kids who during class Don't bother trying to get the same answer as me on their calculator Because this is when you should be making your stupid calculator mistakes When it doesn't count and then on the test, you know what you're doing That's my way of saying you should all be typing these in and seeing that you know Especially as soon as we get to the more complicated stuff You want to get good at your calculators. It drives me crazy the number of kids that have all the right stuff written Wrong answer What do we get Tyson? 2.55 anybody else? Yeah, I got naughty 2.55 seconds to the top. Oh, and how long was it in the air for grand total? What do we say? It's highlighted 6.88. Oh, so you also now know how long was falling down for 6.88 minus 2.5 if I asked that I mean there's all sorts of stuff we could do now What's the key for dealing with a cliff components? Displacement is negative and you're either going to end up using the quadratic equation or if you find VY final It's negative two because it's on its way down Next page is it turn the page for you guys or is it next page over so you still have all the data on your page Good good good good good good. These are the other questions. They like to ask see find the range When they say find the range what they're really saying is find DX The horizontal displacement so in my horizontal column. I have a V T 6.88 and they want me to find D. What's an equation that has those four things in it? So be no louder D equals V I and I'll put X's on everything T plus one half AX T squared and There's something really cool. You look look look look look look. Why can I do that? Because the horizontal acceleration is zero most common mistake kids want to put a negative 9.8 in there because they're like What's the accelerate on 9.8? No not horizontally in Fact the range is just going to be Vx which was 43 something. What was it? 43.3 times 6.88 That one hopefully you don't you can get on your calculator times 3.3. Hey, I got the 6.88 store to my calculator. That's convenient 209 oh three six picks 298 units meters the last one they'll ask and this is fairly challenging is The impact velocity although sometimes they'll say find the final velocity now often when they say that kids go Zero stupid. No, no not after it hits the ground final velocity right when it hits the ground Okay, I'm going to argue that as a picture The final velocity looks like this. I'm gonna argue Kirsten. It's coming at a pretty steep angle Not vertical though because it's still moving sideways that makes sense launching from a cliff I think you'd hit that kind of a steep angle Not only that Christian, I'm gonna argue that we can break this up Into that and that so far so good Where this is my horizontal velocity of 43.3 because it never changes and this is the final vertical velocity just before impact Which conveniently I think I figured this out on the previous page when I showed you the second method of finding time Without using the quadratic formula. It forced me to find the vertical velocity the final vertical velocity Just before impact. What was the final vertical? It was something in the 40s as well if I recall sorry 42.44 technically negative except since I've got a vector arrow pointing down I'm gonna ditch the negative it shows up in my diagram and I think this is Impact velocity How can I find the magnitude of the impact velocity? math 8 Pythagoras and you know what I think this here is going to be my angle theta. Oh This actually isn't too bad if you've done the setup ahead of time Now if you didn't know v y final This is where you would have to crunch the numbers and find we v y final, but we already did conveniently It's gonna be the impact squared equals 43.3 squared plus 42.44 squared the impact Equals 43.3 squared plus 42.44 squared Isn't that negative well technically this is negative, but when you square negative what's gonna happen anyhow So you know what the vector arrow has the negative contained in it. So who cares? square root 60.6 meters per second now that's the magnitude Direction this angle here Alex, but not Alex or Alex Alex What trig function opposite adjacent hypotenuse based on that angle there? What's this one? Agree with you What about this one? Which trig function is the a and e on it What you do in Hawaii tan absolutely in fact Alex? I'm gonna say this the angle is going to be the inverse tangent right of opposite Over Adjacent I went straight to writing what we're gonna do on our calculator and remember the symbol for inverse on your Calculators at stupid negative one so can someone go shift tan of that What do we get for an angle by the way? Cassidy are both these two numbers close to each other The angle is gonna be close to 45 degrees because this is nearly isosceles 44 point 44.4 degrees What of what well here's the problem I can't really say south of east now If you said that I wouldn't take marks off, but I'm gonna suggest to you. That's not north That's not south on a map. Yes up is north But in real life up is up Down is down. So if you said South of east I give it to you But here's how we actually write this you would say 44 degrees below The Horizontal That's how you draw an angle it's all from here down. You say it's below the horizontal What would I say if it was from here up? Do you think? Above the horizontal. Yep That's how we label diagrams from the side North and south doesn't quite work and that's how we do cliffs components negative displacement if it's a vertical one Quadratic formula or find v y final negative, but you need to find t somehow Which is the best method probably depends on your calculator to be quite honest next page Example 11 says find the flight time using the quadratic formula Okay Well, you know what I'm gonna say to you find the flight time any way you want. I don't care Andrew cliff Angle is There an angle? Just say yes, then I'm gonna break this into its components I'm never gonna use that 35 once I've done the components and I'm gonna go horizontal and vertical Step-by-step. It's a fair chunk of writing, but it's fairly meticulous systematic if you're careful This diagram is too small for me again Christian. I'm gonna draw a little bit bigger like that 35 Here's vx Here's v y initial and this was 20 degrees Spencer what it would be great if there was some stupid way that I could get to a shortcut for what the heck v y was What was v y? Oh, you're saying that v y initial is 35 Sign 20 and v x is 35 Coast 20 why I actually agree with you. I know it's amazing. I have no idea what this is 35 sign 20 I'll go 11.97 carry an extra sig figure 2 and I'll go 32.89 Now what? horizontal vertical a x 0 a y negative 9.8 Vx 32.89 V y initial I scroll down to 11.97. What's this question asking me to find? Time means they better give me something else. Oh, they gave me the vertical displacement Justin. What's the vertical displacement? negative 20 which means I'm gonna find t here Now I'm gonna pause and say try this next part try finding the time on your own you can either use the quadratic formula or You can find v y final square square root negative and then use t equals vf minus vi over a plus one half T square Money for my quadratic solver Or if you find v y final first I found v y final was negative twenty three point one three And sure enough when I typed in the numbers there I also got negative three point five eight either way got me there this fun if you didn't get that I'll come around and check probably making usually it's a calculator zig mistake projectiles That's the concept of projectiles All we're gonna do for the next few days is look at weird and bizarre trickier questions Like what if the projectile hits a wall vertically along the way and sticks? How high will it stick? Oh? Figure that out how many got the three point five eight. I'm just curious most of you Most of the rest I'm gonna guess calculator glitch So I'll put this back up in a second But homework I already assigned one and two yes So I can now say hey you can do number three as well Five and six. I haven't yet talked about that's finding this velocity Velocity magnitude and direction at a certain time, but I'll give you a hint remember when we found the impact velocity We had to add two velocities together if you want to find the velocity. Let's say after two seconds Ashley It's gonna be your horizontal plus whatever v y is after two seconds The resultant is your velocity after two seconds at an angle So I'll be doing that next class, but I'm always curious to see who can figure it out on their own and impress me Uh Ten is good and 11 is good. We're nearly done I only actually have about two more lessons and I'm done the unit So I'm also going to say to you remember the great big review that I gave out the one that was called the ultimate Kinematics review I'm gonna start to say to you if you finish this homework You can start working on the review and these are the questions that you're now capable of doing one to four six to eight 11 to 17 19 to 28 and 30 and 31. I'll leave that up there for a few minutes in just a second But first Well the lesson is done right click