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Формулы кинематики, и движение тела, брошенного в воздух

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so I'm curious about how much acceleration does a pilot or the pilot and the plane experience when it when they need to take off from an aircraft carrier so I looked up a few statistics on the Internet this right here is a picture of an f-18 Hornet right over here it has a take-off speed of 260 km/h if we want that to be a velocity 260 km/h in this direction if it's taking off from this Nimitz class carrier right over here and I also looked it up and I found the runway length or I should say the catapult length because these planes are don't take off just with their own power they have their own thrusters going they also are catapulted off so they can be really accelerated quickly off of the flight deck of this carrier and the runway length of a Nimitz class carrier is about 80 meters so this is where they take off from this right over here is where they take off from and then they come in and they land over here but I'm curious about the take-off so to do this let's figure out let's well let's just figure out the acceleration and from that we could also figure out how long it takes them to be catapulted off the flight deck so let me get the numbers in one place so the take-off velocity I could say is 260 kilometers per hour so let me write this down so that has to be your final velocity when you're getting off of the plane if you want to be flying so your initial velocity is going to be 0 and once again I'm going to use the convention that the the direction of the vector is implicit positive means going in the direction of takeoff negative would mean going the other way my initial velocity is 0 I'll denote it as a vector right here my final velocity my final velocity over here has to be 260 260 kilometers per hour and I want to convert everything to meters and seconds just so that I can get my use it at least 4 meters so that I can use my runway length in meters let's just do it in meters per second I have a feeling it'll be a little bit easier to understand when we talk about acceleration in those units as well so if we want to convert this into seconds we have we'll put hours in the numerator one hour so it cancels out with this hour is equal to 3600 seconds 3,600 seconds I'll just write 3,600 s and then if we want to convert it to meters we have 1000 meters 1000 meters is equal to 1 is equal to 1 kilometer and this 1 kilometer will cancel out with those kilometers right over there and whenever you're doing any type of this dimensional analysis you should really should see whether it makes sense if I'm going 260 kilometers in an hour I should go much fewer kilometers in a second because the second is so much a shorter amount of time and that's why we're dividing by 3600 if I can go a certain number of kilometers in an hour a second I should be able to go a lot many many many more meters in that same amount of time and that's why we're multiplying by a thousand but you multiply these out the hours cancel out you have the kilometers cancelling out and you have 260 times a thousand divided by 3600 meters per second so let me get my trusty ti-85 out and actually calculate that I have over here so I have 260 260 times 1000 times 1000 divided by divided by 3600 divided by 3600 gets me I'll just run it to 72 because that's about how many significant digits I can assume here 72 meters per second so all I did here is I converted the take-off velocity so this is 72 meters per second this has to be the final velocity after accelerating so let's think about that acceleration could be given that we know that the length of the runway and we're going to assume constant acceleration here just to simplify things a little bit but what does that constant acceleration have to be so let's think a little bit about it the total displacement I'll do that in purple the total displacement is going to be equal to is going to be equal to our average velocity while we're accelerating times the time times the tot the difference in time or the amount of time that takes us to accelerate now what is the average velocity here it's going to be our final velocity plus our initial velocity over 2 it's just the average of the initial and final and we can only do that because we are dealing with a constant acceleration and what is our change in time over here what does it change in time well our change in time is how long does it take us to get to that to get to that velocity or another way to think about it is it is our change in velocity divided by our acceleration if we're trying to get to if we're trying to get to ten meters per second and we're or we're trying to get 10 meters per second faster and we're accelerating it 2 meters per second squared it'll take us five seconds or if you want to see that explicitly written in a formula we know that acceleration is equal to change in velocity over change in time you multiply both sides by change in time and you divide both sides by acceleration so let's do that multiply both sides by change in time and divide by acceleration multiplied by change in time and divide by acceleration and you get that cancels out and then you have that cancelled out and you have change in time is equal to change in velocity divided by acceleration change in velocity divided by acceleration so what's the change in velocity change in velocity so this is going to be change in velocity divided by acceleration change in velocity is the same thing as your final velocity minus your initial velocity all of that divided by acceleration so this delta T part we can rewrite as our final velocity minus our initial velocity minus our initial velocity over acceleration and just doing this simple little derivation here actually gives us a pretty cool result if we just if we just work through this math and I'll try to write a little bigger I see my right is getting my writing is getting smaller our displacement can be expressed as the product of these two things and what's cool about this well let me just write it this way so this is V our final velocity plus our initial velocity times times our final velocity minus our initial velocity all of that over all of that over 2 times our acceleration our assumed constant our assumed constant acceleration and you probably remember from algebra class this is it takes to form a plus B times a minus B and so this is equal to and you can multiply it out if and you can review in our algebra playlist how to multiply out two binomials like this but this numerator right over here I'll write it in blue is going to be equal to our final velocity squared minus our initial velocity squared this is a difference of squares you can factor it out into the sum of the of the two terms times the difference of the two terms so that when you multiply these two out you just get that over there over over two times the acceleration over two times the acceleration now what's really cool here is we were able to derive a formula that just deals with the displacement our final velocity our initial velocity and the acceleration and we know all of those things except for the acceleration we know that our displacement is 80 meters we know that this is 80 meters we know that our final velocity just before we square it we know that our final velocity is 72 meters per second 72 meters per second and we know that our initial velocity is zero meters per second and so we can use all of this information to solve for our acceleration so and you could you might see this formula displacement sometimes called distance if you're just reading the scalar version and really we are thinking only in the scale ever think about the magnitudes of all of these things for the sake of this video we're only dealing in one dimension but sometimes you'll see it written like this sometimes you'll multiply both sides times the 2a and you'll get something like this where you have two times the really the magnitude of the acceleration times the magnitude of the displacement which is the same thing as the distance is equal to the final velocity the magnitude of the final velocity squared what we're assuming this is so the final velocity squared minus minus the initial velocity squared or sometimes in some books it'll be written as to ad is equal to VF squared minus VI squared it seems like the super mysterious thing but it's not that mysterious we just very simply derived it from displacement or if you want to say distance if you're just thinking about the scalar quantity is equal to average velocity times change in time so so far we've just derived ourselves a kind of a neat formula that is often not derived in physics class but let's use it to actually figure out the acceleration that a pilot experiences when they're taking off of a nimitz-class carrier so we have two times the acceleration times the distance that's 80 meters times 80 meters is going to be equal to our final velocity squared what's our final velocity is 72 meters per second so 72 meters per second squared minus our initial velocity so our initial velocity in this situation is just 0 so it's going to be minus 0 squared which is just going to be 0 so we don't even have to write it down and so to solve for acceleration to solve for acceleration you just divide so this is the same thing as 160 meters well let's just divide both sides by 2 times 80 so we get acceleration is equal to is equal to 72 meters per second squared over 2 times 80 over 2 times 80 meters 2 times 80 meters and what we're going to get is and I'll just write this in one color it's going to be 72 divided by 160 times we have in the numerator meters squared over second squared we're squaring the unit's meter squared over second squared and then we're going to we're going to be dividing by meters so times I'll just in blue times one over meters right because we have a meters in the denominator and so what we're going to get is this meter squared divided by meters that's going to cancel out and we get meters per second squared which is cool because that's what acceleration should be in and so let's just get the calculator out to calculate this exact acceleration so we have to take oh sorry this is 72 squared let me let me write that down so this is this is going to be 72 squared don't want to forget about this part right over here 72 squared divided by 160 so we have and we could just use the original number right that we calculated so let's just square that and then divide that by 160 divided by 160 and if we go to two significant digits we get 33 we get our acceleration is our acceleration is equal to 33 meters per second squared and just to give you an idea of how much acceleration that is is if you are in freefall over Earth the the earth and the force of gravity will be accelerating you at you so G is going to be equal to 9.8 meters per second squared so this is accelerating you three times three times more than what earth is making you accelerate if you were to jump off of a cliff or something so another way to think about this is that the the force and we haven't done a lot on force yet we'll talk about this more in more depth is that this pilot would it be experiencing more than three times the force of gravity more than three g's three g's would be about 30 meters per second squared this is more than that so an analogy for how the pilot would feel is when he's you know if this is the chair right here that his his pilot's chair that he's in so this is the chair and he's sitting on the chair let me do my best to draw him sitting on the chair so this is him sitting on the chair flying the plane and this is the pilot the force he would feel or while this thing is accelerating him forward at 33 meters per second squared it would feel very much to him like as if he was lying down on the surface of the planet but he was three times heavier or more than three times heavier or if he if he was lying down or in he's you know if you were lying down like this let's say this is you this is your feet and this is your face this is your hands let me draw your hands right here and if you had if you had essentially two more people stacked above you roughly I'm just giving you the general sense of it that's how it would feel a little bit more than two people that squeezing sensation so his entire body is going to three times heavier than it would if he was just laying down on the beach or something like that so it's very very very interesting I guess idea at least to me now the other question that we can ask ourselves is how long will it take to get catapulted off of this off of this carrier and if he's accelerating at 33 meters per second squared if he's accelerating at 33 meters per second squared how long would it take him to go from 0 to 72 meters per second so after one second he'll be going 33 meters per second after 2 seconds he'll be going 66 meters per second so it's going to take and so it's a little bit more than 2 seconds so it's going to take them a little bit more than 2 seconds and we could calculate it exactly if you take 72 meters per second and you divide it by 33 it'll take them 2.18 seconds roughly 2x2 be catapulted off of that carrier