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# Projectile motion (part 3)

## Video transcript

in the last video I said that we start off with let's say we started off with the change in distance so we said you know that we know the change in distance the change in distance this is the things that we are given we're given the acceleration we're given the initial velocity and I asked you how do we figure out what the final velocity is and in the last video and if you don't remember to go go watch that last video again we derived the formula that VF squared the final velocity squared is equal to the initial velocity squared plus 2 times the change in distance you'll sometimes just see written this as 2 times distance because we assume that the initial distance is at point 0 so the change in distance would just be the confining distance but we could write it either way and hopefully at this point you kind of see why I keep switching between change in distance and distance really just so you're comfortable when you see it either way but what if we this is this is for the situation if when we didn't know what VF is but let's say we want to solve for for time instead well actually once we solve for the final velocity we can actually solve for time and I'll show you how to do that but so we didn't want to go through this step how can we solve for time directly given the change in distance the acceleration and the initial velocity well let's go back once again to our the most basic distance formula but not the distance formula how distant relates to velocity so we know that I'll write it slightly different this time - so the change in distance over the change in time is equal to the average velocity all right or we could if we could have rewritten this is we could rewrite this as the change in distance is equal to the average velocity times the change in time all right this is change in time change in distance and sometimes we will just see this written as d equals let me write this in a different color so we have some variety sometimes you'll see this is written as d equals velocity times time or D equals rate times time and the reason why I have change in distance here or change in time is well I'm not assuming necessarily that we're starting off at the point zero or a time zero but if we do then it just turns out to kind of the final distance is equal to the average velocity times the final time but let's stick to this we want to figure out time given this set of inputs so let's go back to well let's let's go let's go back to this equation actually let's go from this equation right so if we want to solve for time or the change in time we could say we could just drive both side we could divide both sides by the average velocity actually no let's not do that let's just stay in terms of change in distance so let me let me actually I've wasted space too fast so let me clear this and start again so we're given we're given change in distance we're given initial velocity that's initial velocity and we're given acceleration and we want to figure out what the time is and if we assume I mean it's really the change in time but let's just assume that we start at time zero so this kind of final time so we know that let's just start with a simple formula distance or change in distance I'll use them interchangeably distance a little our case D this time is equal to the average velocity times time and what's the average velocity well that is just the average velocity is just the initial velocity plus the final velocity over two and the only reason why we can do it why we can just average the initial in the finalists because we're assuming constant acceleration that's very important but in most projectile problems we do have constant acceleration downwards and that's gravity so we can ASSU do this we consider the average of the initial and the final velocity is the average velocity and then we multiply that times and well can we use this equation directly well no we know acceleration but we don't know final velocity so if we can write this final velocity in terms of the other things in this equation then maybe we can we can solve for time let's try to do that so when I switch colors kind of arbitrarily but let's distance is equal to well let me let me take a little side here because what do we know about final velocity we know that the change in velocity the change in velocity is equal to acceleration times time right assuming the time starts at T equals zero and the change in velocity is the same thing as VF minus VI is equal to acceleration times time and so we know the final velocity is equal to the initial velocity plus acceleration times time all right so let's stop to do that back into this what I was writing right here so we have distance is equal to the initial velocity plus the final velocity so let's substitute this expression right here the initial velocities plus now the final velocity is the initial velocity plus acceleration times time and then we divide all of that by two times time and so we get D is equal to so let's see we have two in the numerator we have two the initial velocity - V is plus a T over two all of that times T and then we can simplify this this equals D is equal to let's see this to cancel out this two so we have and we multiply it we distribute this T across both terms so D is equal to V I T plus and then this term is a T over two but then you multiply the T times here - so it's a T squared over two plus a T squared over two so there we could use this formula if we know the change in distance or the distance we could say you know we could say the same thing is the change in this this actually should be the change in distance and the change in time is equal to the initial velocity times time plus acceleration times squared divided by two so let me summarize kind of all of the equations we have because we really now have in our arsenal every equation that we that you really need to solve one dimensional projectile problem you know things going either just left-right east-west or north-south and not both and we'll do that in the next video but let's let's summarize everything we know image so we know the change in distance divided by the change in time is equal to velocity average velocity it would equal velocity velocity is not changing but average wind velocity does change and we have constant acceleration that's an important assumption we know that the change in velocity divided by the change in time is equal to acceleration we know the average velocity is equal to the final velocity plus the initial velocity over two and this assumes acceleration is constant acceleration constant if we know the initial velocity the acceleration and the distance and we want to figure out we want to figure out the final velocity we could go we could use this formula V F squared equals VI squared plus 2ei times really the change in distance I'm going to write the change in distance because that sometimes matter is when we're when we're dealing with the direction times change in distance but you'll sometimes just written this as distance and then we just did the equation I think I did this in the third video as well early on but we also learned that distance is equal to the initial velocity times time plus a squared over 2 so in that example that I did a couple of videos ago where we had a cliff actually I only have a minute left in this video so I'm going to do that in the next presentation I'll see you soon