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Current time:0:00Total duration:9:58

Deriving displacement as a function of time, acceleration, and initial velocity

AP.PHYS:
CHA‑4.A (EU)
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CHA‑4.A.1 (EK)
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CHA‑4.A.1.1 (LO)
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CHA‑4.A.2 (EK)
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CHA‑4.A.2.1 (LO)
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CHA‑4.A.2.3 (LO)

Video transcript

what I want to do in this video is think a little bit about what happens to some type of projectile maybe a ball a ball or rock if I were to throw it up straight up into the air so to do that and what I want to do is on a plot its distance relative to time so there's a few things that I'm going to tell you about my throwing of the rock in the air well I'll have an initial velocity I'll have an initial velocity of nineteen point six meters per second and I picked this initial velocity because it'll make the math a little bit a little bit easier and we also know the acceleration near the surface of the earth we know that the force of gravity near the surface of the earth is is the mass of that object times let me write this down the force of gravity the force of gravity on an object near the surface of the earth is going to be the mass of the object times little G times gravity on earth this is 9.8 meters meter meters per second squared now if you want the acceleration on earth you just divide force divided by mass right because we have where force is equal to mass times acceleration if you want acceleration divide both sides by mass so you get force over mass so let's just divide this by mass let me divide both sides of this by mass if you divide both sides of this by mass on this on the left hand side you get acceleration and on the right hand side right over here you get the same quantity G and the whole reason why I did this is when we look at the little G it really comes from the universal law of gravitation and you can really view it as the as measuring the field strength the gravitational field strength near the surface of the earth and then that helps us figure out when you multiply it by mass the force then you use F equals MA Newton's second law to come up with G again which is actually the acceleration and accelerating you towards the center of the earth the other thing I want to make clear and you might have already thought about this is when you talk about gravity the force of gravity generally you're saying look the force of gravity is equal to big G which is different than the little G times the product of the masses of the two things times the square of the distance between the two objects and so you might be saying wait clearly clearly the force of gravity is dependent on the distance and so if I were to throw something up into the air well the distance change and you would be right that is technically right but the reality is is that when you throw something up in the air that change in distance is so small relative to the distance between that object and the center of the earth that to make the math simple when we're at the when we're at or near the surface of the earth including in our atmosphere we can assume it is constant we can assume it is constant remember that little G right over there is really all of these terms combined if we assume that mass one is the scent is the earth or mass of the earth mass of the earth and our right here is the distance really the radius of the earth from the surface of the earth to the center of earth radius radius of Earth so you'd be correct in thinking that it actually changes a little bit the force of gravity changes a little bit but for the sake of throwing things up in our atmosphere and all that we can assume that it is constant and if you were to calculate it it is 9.8 meters per second squared and I've rounded here to the nearest tenth and I want to be clear these are vector quantities and in this when we start throwing stuff up in the air the convention is is that up if something is moving up it is we will give it a positive value and if something is moving down we give it a negative value well gravity would be except for an object that's in freefall gravity would be accelerating it downwards or the force of gravity is downwards so little G right over here if we want to give its direction is negative is negative 9.8 meters per second squared so we have the acceleration due to gravity so the acceleration due to gravity is going to be negative 9.8 meters per second squared and I want to plot distance relative to time so let's think about let's think about how we can set up a formula how we can derive a formula that if we input time as a dependent variable we can get distance and we could we can assume these values right over here well we know or even more important I want to plot displacement versus time because that'll actually be more interesting in this problem so we know that displacement we know that displacement is the same thing as average velocity average velocity times times change in time so right now we have something in terms of time distance and average velocity but not in terms of initial velocity and acceleration right over here but we know that average velocity we know that average velocity is the same thing as initial velocity plus final velocity over two if we assume constant acceleration so we can we can do this assuming assuming constant acceleration and once again and once again when we're dealing we're not too far from the surface of the earth we can make that assumption assuming we have a constant acceleration but this right here once again we don't have what our final velocity is so we need to think about this a little bit more but we can express our final velocity in terms of our initial velocity and time so this expression right here so we're just dealing with this part of the average velocity so we can rewrite this expression as the initial velocity plus something over two and what is final velocity well the final velocity I'll do it in that same yellow color the final velocity is going to be your initial velocity plus your plus your acceleration plus your acceleration times change in time right if you're starting at ten meters per second and you were accelerated one meters per second squared then after a second you'll be going one meter per second faster than that so this right here this right here is your is your final velocity let me make sure that these are all vector quantities so all of these things over here are vector quantities I've sometimes might forget that forgive me I hope hopefully by now it's pretty ingrained in you that these are all vector quantities the direction matters but let's see if how we can simplify this well these two terms remember we're just dealing with this average velocity term right over here these two term these two if you combine them become two VI two times my initial velocity and then divided by this two divided by this two plus all of this business divided by this two plus my acceleration times change in time divided by two so all of this was another way to write average velocity and the whole reason why I did this is because we don't have final velocity here but we have acceleration and we're going to use change in time as our independent variable and so we also have to still multiply this by this green change in time right over here by this green change in time and all of this was what displacement is going to be this is displacement and let's see we can multiply the change in time times all of this business and actually these twos cancel out and we get I'll continue it over here so I can use all of the real estate we get the displacement displacement is equal to the initial velocity is equal to initial velocity times change in time times change in during that same color times change in time and a lot of times in a lot of physics books or tree or traditional physics classes they'll just write time there but it's really change in time I mean you know it's not super wrong if there's just a tea here but change in time is a little bit more accurate and then plus plus you have 1/2 that's the same thing as dividing by 2 plus 1/2 times the acceleration times the acceleration times we have a delta t times delta t change in time times change in time this triangle just means change in so change in time times change in time is just change in time squared change in time squared and in some classes you will sometimes see this written as you know maybe D is equal to VI times T plus one-half a t-square same exact thing they're just you d as the variable for displacement and they're using T instead of delta T these are the same things but the one thing I want you to realize in this video is that this is just a very this is a very straightforward thing to derive you really may be if you're under time pressure you might want to just make sure that you know just be able to whip this out but the important thing so that you remember how to do this when you're 30 years old or 40 years old or 50 years old or when you're an engineer you're trying to send a rocket into space and you don't have a physics book to look up is it just comes from the simple displacement is equal to average velocity times change in time Andreea soom constant acceleration and then you can kind of just derive the rest of this now I'm going to leave you there in this video we're going to we're going to leave off let me erase this part right over here I'm going to leave it right over here in the next video we're going to use this formula we just derived we're going to use this to actually plot to plot the displacement versus time because that's interesting then we're going to be thinking about what happens to the velocity and the acceleration as we move as as as we move further and further ahead in time