Learn about Projectile motion - part 3 Video

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Learn about Projectile motion - part 3

In the last video, I said that we start off with the change in distance, so we solve the other. We would know the change in distance. These are 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, go watch that last video again. We derived the formula that vf2, the final velocity squared is equal to the initial velocity squared plus two times the change in distance. And sometimes you’ll see written this as two times distance because we assumed that the initial distance is at point zero, so the change in distance would just be the final distance, but we can 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. This is for the situation when we didn’t know what vf is, but let’s say we want to solve 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. Let’s say we didn’t want to go through this step, how can we solve for time directly given the change in distance the acceleration of the initial velocity? Well, let’s go back once again to the most basic distance formula, not the distance formula, how distance relates to velocity. So, we know that it’s slightly different this time so the change in distance over the change in time is equal to the average velocity alright or we could have rewritten this as—we could rewrite this as the change in distance is equal to the average velocity times the change in time. Alright, this is change in time, change in distance. And sometimes we all just see this written as d = velocity times time or d = 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 at 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 this equation or actually just go from this equation, right? So, if we want to solve for time or the change in time, we could say we can divide both sides by the average velocity. Actually, let’s not do that. Let’s just stay in terms of change in distance. So, we’re given change in distance, we’re given initial velocity, its initial velocity and we’re given acceleration. And we want to figure out what the time is and if we assume just really the change in time but let’s just assume that we start time zero so this is kind of the final time. Let’s just start with this simple formula, distance or change in distance, if I use them interchangeably, distance is equal to the average velocity times time. And what’s the average velocity? The average velocity is just the initial velocity plus the final velocity over two. And the only reason why we can do it while we can just average the initial and the finals 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 do this. We can say that the average of the initial and the final velocity is the average velocity and then we multiply that times time. 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 solve for time. Well, let’s try to do that, so I switch colors kind of arbitrarily but distance is equal to—well, let me take a little side here because what do we know about final velocity? We know that the change in velocity is equal to acceleration times time. Alright, assuming that times starts at T equals zero. And the change in velocity is the same thing as vf - vi is equal to acceleration times time. And so we know the final velocity is equal to the initial velocity plus acceleration times time. Alright, so let’s substitute 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 velocities 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 so let’s see we have two and the numerator we have two initial velocity, two vi’s plus AT over 2, all of that times T and then we can simplify this. This equals d is equal to let’s see this two cancel out this two and we distribute this T across both terms. So D is equal to VIT plus and then this term is AT over two, but then you multiply the T times here 2 so it’s AT squared over two plus AT squared over two, so there. We could use this formula if we know the change in distance or the distance. We could say that’s the same thing as the change in distance. This is 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 the equations we have because we really now have an answer, every equation 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 videos. But let’s summarize everything we know. So, we know the change in distance divided by the change in time is equal to velocity, average velocity. It would equal velocity if velocity is not changing but average when 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 as constant. If we know the initial velocity, the acceleration and the distance and we want to figure out the final velocity, we could use this formula vf2 = vi2 + 2a times really the change in distance. So, I'm going to write the change in distance because that sometimes matters when we’re dealing with the direction times change in distance but you’ll sometimes just write 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 learn that distance is equal to the initial velocity times time plus AT squared over two. So, in that example that I did a couple of videos ago where we had a clip, 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.

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In the last video, I said that we start off with the change in distance, so we solve the other. We would know the change in distance. These are 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, go watch that last video again. We derived the formula that vf2, the final velocity squared is equal to the initial velocity squared plus two times the change in distance. And sometimes you’ll see written this as two times distance because we assumed that the initial distance is at point zero, so the change in distance would just be the final distance, but we can 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.

This is for the situation when we didn’t know what vf is, but let’s say we want to solve 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. Let’s say we didn’t want to go through this step, how can we solve for time directly given the change in distance the acceleration of the initial velocity? Well, let’s go back once again to the most basic distance formula, not the distance formula, how distance relates to velocity. So, we know that it’s slightly different this time so the change in distance over the change in time is equal to the average velocity alright or we could have rewritten this as—we could rewrite this as the change in distance is equal to the average velocity times the change in time.

Alright, this is change in time, change in distance. And sometimes we all just see this written as d = velocity times time or d = 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 at 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 this equation or actually just go from this equation, right? So, if we want to solve for time or the change in time, we could say we can divide both sides by the average velocity. Actually, let’s not do that. Let’s just stay in terms of change in distance. So, we’re given change in distance, we’re given initial velocity, its initial velocity and we’re given acceleration. And we want to figure out what the time is and if we assume just really the change in time but let’s just assume that we start time zero so this is kind of the final time.

Let’s just start with this simple formula, distance or change in distance, if I use them interchangeably, distance is equal to the average velocity times time. And what’s the average velocity? The average velocity is just the initial velocity plus the final velocity over two. And the only reason why we can do it while we can just average the
initial and the finals 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 do this. We can say that the average of the initial and the final velocity is the average velocity and then we multiply that times time. 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 solve for time. Well, let’s try to do that, so I switch colors kind of arbitrarily but distance is equal to—well, let me take a little side here because what do we know about final velocity? We know that the change in velocity is equal to acceleration times time. Alright, assuming that times starts at T equals zero. And the change in velocity is the same thing as vf - vi is equal to acceleration times time. And so we know the final velocity is equal to the initial velocity plus acceleration times time.

Alright, so let’s substitute 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 velocities 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 so let’s see we have two and the numerator we have two initial velocity, two vi’s plus AT over 2, all of that times T and then we can simplify this. This equals d is equal to let’s see this two cancel out this two and we distribute this T across both terms. So D is equal to VIT plus and then this term is AT over two, but then you multiply the T times here 2 so it’s AT squared over two plus AT squared over two, so there. We could use this formula if we know the change in distance or the distance. We could say that’s the same thing as the change in distance. This is 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 the equations we have because we really now have an answer, every equation 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 videos. But let’s summarize everything we know. So, we know the change in distance divided by the change in time is equal to velocity, average velocity. It would equal velocity if velocity is not changing but average when 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 as constant. If we know the initial velocity, the acceleration and the distance and we want to figure out the final velocity, we could use this formula vf2 = vi2 + 2a times really the change in distance. So, I'm going to write the change in distance because that sometimes matters when we’re dealing with the direction times change in distance but you’ll sometimes just write 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 learn that distance is equal to the initial velocity times time plus AT squared over two.

So, in that example that I did a couple of videos ago where we had a clip, 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.

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