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# Horizontally launched projectile

## Video transcript

let's talk about how to handle a horizontally launched projectile problem these technically speaking if you already know how to do projectile problems there's nothing new except that there's one aspect to these problems that people get stumped by all the times I'm going to show you what that is in a minute so that you don't fall into the same trap what we mean by a horizontally launched projectile is any object that gets launched in a completely horizontal velocity to start with so if something is launched off of a cliff let's say in this straight horizontal direction with no vertical component to start with then it's a horizontally launched projectile what could that be I mean a boring example is just a ball rolling off of a table if you just roll the ball off of a table then the velocity the ball has to start off with if the table is flat and horizontal the velocity of the ball initially would just be horizontal so if the initial velocity of the object for a projectile is completely horizontal then that object is a horizontally launched horizontally launched projectile a more exciting example people do crazy stuff let's say this person is going to cliff dive or base jump and they're going to be like whoa let's do this we're going to do this they're pumped up they're going to run but they don't jump off the cliff edge of the run straight off of the cliff because they're kind of nervous let's say they run off of this cliff with 5 meters per second of initial velocity straight off the cliff and let's say they're completely crazy let's say this cliff is 30 meters tall so that's like over 90 feet that is kind of crazy so 30 meters tall they launch they fly through the air there's water down here so they initially went this way and they start to fall down and they do something like and then they splash into the water hopefully they don't hit any boats or fish down here that fish already looks like he got hit he or she all right fish over here person splashes in the water we want to know here's a question you might get asked how far did this person go horizontally before striking the water this is a classic problem it's asked all the time and if you were a cliff diver I mean don't try this at home but if you're a professional cliff diver you might want to know this cliff height and the speed how fast do I have to run in order to avoid maybe the rocky like sure right here that you might want to avoid maybe there's a nasty craggy cliff bottom here that you can't fall on so how fast would I have to run in order to make it past that alright so conceptually what's happening here same thing that happens for any projectile problem the horizontal direction is happening independently of the vertical direction and what I mean by that is that the horizontal velocity evolves independent to the vertical velocity let me give the velocity this color let's say the vertical velocity or the vertical direction of pink horizontal direction is green this vertical velocity is going to be changing but this horizontal velocity just going to remain the same these do not influence each other in other words this horizontal velocity started at five the person is always going to have five meters per second of horizontal velocity so this horizontal velocity is always going to be five meters per second the whole trip assuming this person really is a freely flying projectile assuming that there was no jetpack that propelled them forward or no air resistance this person is always going to have five meters per second of horizontal velocity up into the point right and when they splash into the water and then at that point there's forces from the water that that influence this acceleration in various ways that we're not going to consider how about vertically vertically this person starts with no initial velocity so this person just ran horizontally straight off the cliff and then they start to gain velocity so they're going to gain vertical velocity downward and maybe more vertical velocity because gravity keeps pulling and then even more this might go off the screen but it's going to be really big so a lot of vertical boxes this should keep getting bigger and bigger and bigger because gravity's influencing this vertical direction but not the horizontal direction so how do we solve this with math let's write down what we know what we know is that horizontally this person started off with an initial velocity V initial in the X I could have wrote I for initial but I wrote zero for V naught in the X it still means initial velocity is five meters per second and we don't know anything else in the X direction you might think thirty meters is the displacement in the extra affection but that's a vertical distance this is not telling us anything about this horizontal distance this horizontal distance or displacement is what we want to know this horizontal displacement in the X direction that's what we want to solve for so we're going to declare our ignorant write that here we don't know how to find it but we want to know that we do want to find this I'm going to write it there how about in the Y direction what do we know we know that the alright now we can use this 30 you might want to say that Delta Y is positive 30 but you would be wrong and the reason is this person fell downward 30 meters think about it they started at the top of the cliff ended at the bottom of the cliff that means this person is going to end up below where they started 30 meters below where they started so this has to be negative 30 meters for the displacement assuming you're treating downward is negative which is typically the convention chosen that downward is negative and leftward is negative so if you choose downward is negative this has to be a negative displacement what else do we know vertically well for a freely freely flying object excuse me we know that the acceleration vertically is always going to be negative 9.8 meters per second squared assuming downward is negative now here's the point where people get stumped here's the part when people make a mistake they want to say that the initial velocity in the Y Direction is 5 meters per second I mean people are just dying to stick this 5 meters per second into here because that's a velocity that you were given but this was a horizontal velocity so why this is called horizontally launched projectile motion not vertically launched projectile motion so think about it the initial velocity in the vertical direction here was 0 there was no initial vertical velocity this person was not launched vertically up or vertically down this person was just launched straight horizontally and so the initial velocity in the vertical direction is just zero people don't like that they're like hold on a minute they're like this person I'm going to start gaining right this person is going to start gaining velocity right when they leave the cliff that starts getting bigger and bigger and bigger in the downward direction but that's after you leave the cliff we're talking about right as you leave the cliff that moment you left the cliff there was only horizontal velocity which means you started with no initial vertical velocity so this is the part people get confused by because this is not given to you explicitly in the problem the problem won't say find the distance for a cliff diver assuming the initial velocity in the y-direction was zero now they're just going to say a cliff diver ran horizontally off of a cliff find this stuff and you're just going to have to know that okay if I run off of a cliff horizontally or something gets shot horizontally that means there was no vertical velocity to start with I'm going to have to plug this initial velocity in the y-direction is zero so that's the trick don't fall for it now you know how to deal with it so we want to solve for displacement in the x-direction look at how many variables we know in the y-direction I mean we know all this this is good but we can't use this to solve directly for the displacement in the X direction we need to use this to solve for the time because the time is going to be the same for the X direction and the y direction so once I find the time I can plug back in over to here because think about it the time it takes for this trip is going to be the time it takes for this trip it doesn't matter whether I call it the X direction or Y Direction time is the same for both the directions in other words the time it takes for this displacement of negative 30 is going to be the time it takes for this displacement of whatever this is that we're going to find so let's solve for the time now how will we do that think about it we know the displacement we know the acceleration we know the initial velocity and we know the time note that we don't know the final velocity and we're not asked to find the final velocity we don't want to know it so let's use a formula that doesn't involve the final velocity and that would look like this so if we use Delta y equals V initial in the Y Direction times time plus 1/2 acceleration in the Y Direction times time squared all right now we can plug in values my displacement in the Y Direction is negative 30 my initial velocity in the Y Direction is zero here's this is where it would happen this is where the mistake would happen people would people just really want to plug that v in over here but don't do it it's a trap so 0 times T is just zero so that whole term is zero plus one half the acceleration is negative 9.8 meters per second squared and then times T squared all right now I can solve for T I'm going to solve for T and then I'd have to take a square root of both sides because it's T squared and what would I get I'd have to multiply both sides by two so I get negative 30 meters times two and then I have to divide both sides by negative 9.8 meters per second squared equals notice if you would have forgot if you would have forgot this negative up here for negative 30 you to come down here this would be a positive up top you would had a negative on the bottom you to plug this in you would have tried to take a square root of a negative number your calculator would have been all like I don't know what that means and you would have been like oh I messed up so you'd start coming back here probably be like let's just make stuff positive and see if that works it would work because look at these negatives cancel but it's best to just know what you're talking about in the first place so be careful plug in your negatives and things will work out all right so if you solve this you get the time it took is two point four seven seconds it's actually a long time might seem like you're falling for a long time sometimes when you're like jumping off of a table jumping off a trampoline that's usually like a fraction of a second this is actually a long time two and a half seconds or free Falls a long time so we could take this that's how long it took to displace by 30 meters vertically but that's going to be how long it took to displace this horizontal direction we can use the same formula we can say that well if Delta x equals V initial in the X direction I'm just using the same formula but in the X Direction plus one half ax T squared so the same formula is this just in the X direction well Delta X is just DX we already gave that a name so let's just call this DX so I'm going to scooch this equation over here DX is Delta X that equals the initial velocity in the X direction that's five all right this is really five now in the x-direction the initial velocity really was five meters per second how about the initial time or sorry the time there's no initial time the time here was two point four seven seconds this was the time interval the time between when the person jumped or ran off the cliff and when the person splashed in the water was two point four let me erase this two point four seven seconds so two point four seven seconds and this comes over here how about this ax this ax is zero remember there's nothing compelling this person to start accelerating the x-direction think if they've got no jet pack there's no air resistance there's no reason this person's going to accelerate horizontally they maintain the same velocity - the whole way so what do we get if we solve this for DX we'd get the DX is about twelve point four I believe let's see I calculated this twelve point four ish meters okay so if these rocks these rocks down here extend more than 12 meters you definitely don't want to do this I mean if it's even close you probably wouldn't want to do this in fact just for safety don't don't try this at home leave this to professional cliff divers I'm just saying if you were one and you wanted to calculate how far you'd make it this is how you would do it so long story short the way you do this problem and the mistakes you would want to avoid are make sure you plug in your negative displacement because you fail downward but the big one is make sure you know that the horizontal velocity or sorry the initial vertical velocity is zero because there was only horizontal velocity to start with that's not going to be given explicitly you're just going to have to provide that on your own and your own knowledge of physics