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this right here is a picture of an Airbus a380 aircraft and I was curious how long would it take this aircraft to take off and I looked up its takeoff velocity so take take off velocity and the specs I got were 280 kilometers per hour and to make this a velocity we have to specify a direction as well not just a magnitude so the direction is in the direction of the runway this that would be the positive direction right over there so we're when we're talking about acceleration or velocity and this we're going to assume it's in this direction the direction of going down the runway and I also looked up its specs and this I'm simplifying it a little bit because it's not going to have a purely constant acceleration but let's just say from the moment that the pilot says we're taking off to when it actually takes off it has a constant acceleration its engines are able to provide a constant acceleration acceleration acceleration of 1.0 m/s per second so after every second it can go one meter per second faster than it was due going at the beginning of that second or another way to write this is 1.0 1.0 m/s per I'd it this way meters per second per second can also be written as meters per second squared I find this a little bit more intuitive this is a little bit neater to write so let's figure this out so the first thing that I so we're trying to answer is how long how long does takeoff last take off how long does takeoff last that is the question we will try to answer and to answer this at least my brain wants to at least get the unit's right so over here we have our acceleration in terms of meters and seconds or second squared and over here we have our takeoff velocity in terms of kilowatt kilometers and hours so let's just convert this takeoff velocity into meters per second and then it might simplify this answering this question so if we have 280 kilometers per hour how do we convert that to meters per second so let's convert it to kilometers per second first so we want to get rid of this hours and the best way to do that if we have an hour in the denominator we want an hour in the numerator and we want it we want a second in the denominator and so how do what what do we multiply this by or what are what do we put in front of the hours in seconds so one hour in one hour there are 3600 seconds 60 seconds in a minute 60 minutes in an hour and so you have one of the larger unit is equal to 3600 of the smaller unit and so we can multiply by that and if we do that the hours will cancel out and we'll get 280 divided by 3600 kilometers per second but I want to do all my math at once so let's also do the conversion from kilometers to meters so once again we have we have kilometers in the numerator so we want the kilometers in the denominator now so it cancels out and we want meters in the numerator and what's the smaller unit it's meters and we have 1,000 meters for every one kilometer and so when you multiply this out the kilometers are going to cancel out and you're going to be left with 280 280 times 1 so we don't have to write it down times 1000 times 1000 all of that over 3,600 all of that over 3,600 and the unit's we have left are meters we have meters per and the only unit we have left here is second meters per second so let's get my trusty ti-85 out and actually calculate this so we have 280 times 1,000 which is obviously 280,000 but let me just divide that by 3,600 3,600 and it gives me 77.7 repeating indefinitely and it looks like I had two significant digits in each of these original things I had 1.0 over here not 100% clear how many significant digits I over here did they did was the spec rounded to the nearest 10 kilometers or is it exactly 280 kilometers per hour just to be safe I'll assume that it's rounded to the nearest 10 kilometers so we only have two significant digits here so we should only have two significant digits in our answer so we're going to round this to 78 78 meters per second so this is going to be this is going to be 78 meters per second which is pretty fast in every symbol for this thing to take off every second that goes by it has to travel 78 meters almost three roughly 3/4 the length of a football field in every second but that's not what we're trying to answer we're trying to say how long will takeoff last well we could just do this in our head if you think about it the acceleration is one meter per second per second which tells us at after every second it's going one meter per second faster so if you start at a velocity of zero and then after one second we'll be going one meter per second after two seconds we'll be going two meters per second after three seconds I'll be going three meters per second so how long will it take to get to 78 meters per second well it'll take 78 seconds it will take 78 it'll take 78 seconds or roughly a minute and 18 seconds and just to verify this with our our definition of our acceleration so to speak just remember acceleration which is a vector quantity in all the directions we're talking about an hour in the direction of this direction of the runway the acceleration is Igor acceleration is equal to change in velocity change in velocity over change in time over change in time and we're trying to solve for how much time does it take or the change in time so let's do that so let's multiply both sides by change in time you get change in time times acceleration times acceleration is equal to change in velocity is equal to change in velocity and to solve for change in time divide both sides by the acceleration so divide both sides by the acceleration you get change in time J I could go down here but I just want to use all this real estate I over here have change in time is equal to change in velocity change in velocity divided by acceleration divided by acceleration and in this situation what is our change in velocity well we're starting off with the velocity or assuming we're starting off with the velocity of zero meters per second and we're getting up to 78 meters per second so our change in velocity is the 78 meters per second so our change in velocity is it is 70 so this is equal in our situation seventy eight meters per second is our change in velocity I'm taking the final velocity 78 meters per second and subtract from that the initial velocity which is zero meters per second and you just get this divided by the acceleration divided by one meter per second per second or one meter per second squared so the numbers part are pretty easy you have 78 divided by 1 which is just 78 and then the unit's you have meters per second and then if you divide by meters per second squared that's the same thing as multiplying by second squared per meter right divided dividing by something is the same thing as multiplying by its reciprocal and you can do the same thing with units and then we see the meters cancel out and then second squared divided by seconds you're left just left with seconds so once again we get 78 seconds a little over a minute for this thing to take off