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# Solving for time

AP.PHYS:
INT‑3.A (EU)
,
INT‑3.A.1 (EK)
,
INT‑3.A.1.1 (LO)
,
INT‑3.A.1.2 (LO)
,
INT‑3.A.1.3 (LO)

## Video transcript

let's work through another few scenarios involving displacement velocity and time or distance rate and time so over here we have then is running at a constant velocity of three minutes or three meters per second to the east three meters per second to the east and just as a review this is a vector quantity they're giving us the magnitude and the direction if they just said three meters per second then that would just be speed so this is the magnitude magnitude is three meters per second and it is to the east so they are giving us the direction so this is a vector quantity and that's why it's velocity instead of speed how long will it take him to travel 720 how long will it take him to travel 720 meters so let's just remind ourselves a few things and I'll do it both with the vector version of it and and maybe they should say how long will take them to travel 720 meters to the east two to the east to make sure to make it clear that it is a vector quantity because so that it's displacement as opposed to just distance but we'll do it both ways so one way to think about it if we think about just the scalar version of it we said already that rate or speed is equal to the distance that you travel over some time over I might write T there but it's really a change in time so sometimes some people would write a little triangle a delta there which means change in time but that's implicitly meant when you just write over time like that so rate or speed is equal to distance divided by time now if you know they're giving us in this problem they're giving us the rate if we think about the scalar part of it they're giving us the rate they're telling us that that is three meters per second and they are also telling us the time or sorry they're not telling us the time they're they're telling us the distance and they want us to figure out the time so they tell us the distance is 720 meters 720 meters and so we just have to figure out the time so we have so if we just do the scalar version of it we're not dealing with velocity and displacement we're dealing with rate or speed and distance so we have three meters per second is equal to 720 meters 720 meters over some change in time over some change in time and so we can algebraically manipulate this we can multiply both sides times time multiply both sides times time multiply time right over there and then we could in if we all if we wanted well let's just take it one step at a time so three meters per second three meters per second times time times time is equal to 720 meters is equal to 720 meters because the times on the right will cancel out right over there and that makes sense at least units wise because time is going to be in seconds seconds cancel out with the seconds in the denominator so you'll just get meters so that just makes sense there so if you want to solve for time you can divide both sides by 3 meters per second so divide both sides by 3 meters per second 3 meters per second and then the left side they cancel out on the right-hand side this is going to be equal to 720 divided by 3 divided by 3 times meters that's meters in the numerator and you had meters per second in the denominator if you bring it out to the numerator you take the inverse of this so that's meters let me do the the meters that was on top let me do that in green let me color code it so 720 meters and now you're dividing by meters per second that's the same thing as multiplying by the inverse times seconds per meters and so what you're going to get here the meters are going to cancel out and you're going to get 720 divided by 3 seconds so what is that 720 divided by 3 72 divided by 3 is 24 so this is going to be 240 this part right over here is going to be 240 and it's going to be 240 seconds that's the only unit we're left with and on the left hand side on the left hand side we just had the time so the time is 200 and 40 seconds sometimes you'll see it and just to show you you know in some business classes they'll show you all these formulas but one thing I really want you to understand is we go through this this journey together is that all those formulas are really just algebraic manipulations of each other so you really shouldn't memorize any of them you should always say hey that's just manipulating one of those other formulas that I got before one of those and even these formulas are hopefully reasonably common sense and so you can take start from very common sense things rate is distance divided by time and then just manipulate it to get other hopefully common sense things so we could have done it here so we could have multiplied both sides by time before we even put in the variables and you would have gotten so if you multiplied both sides by time here by time here you would have got on the right hand on the right hand side distance is equal to time times rate or rate times time and this is one of you'll often see this is kind of the the formula for rate or the formula for motion so if we if we flip it around you get distances equal to rate times time so these are all saying the same things and then if you wanted to solve for time you could divide both sides by rate and you get distance divided by rate is equal to time and that's exactly what we got distance divided by rate was equal to time so if your distance is 720 meters your rate is 3 meters per second 720 meters divided by 3 meters per second will also give you a time of 240 seconds if we wanted to do the exact same thing but the vector version of it just the notation will look a little bit different and we want to keep track of the actual direction so we could say we know that velocity velocity and it is a vector quantity so I put a little arrow on top velocity is the same thing as displacement as let me pick a nice color for displacement blue as displacement and remember we use s for displacement you know we don't want to use D because when you start doing calculus especially vector calculus well any type of calculus you use D for the derivative operator if you don't know what that is don't worry about it right now but this right here is displacement this placement at least this is convention you could kind of use anything but this is what most people use so if you don't want to get confused or if you don't want to be confused they use s it's good to practice with it so it's a displacement per time so it's displacement divided by time sometimes once again you'll have displacement per change in time which is really a little bit more correct but I'll just go with the time right here because this is the convention that you see at least in most beginning physics books so once again if we want to solve for time you can multiply both sides by time and you get this cancels out you get and I'll flip this around well actually I'll leave it like this so you get displacement is equal to is equal to I can flip these around velocity velocity times change in time I should say or we could just say time just to keep things simple and if you want to solve for time you divide both sides by velocity you divide both sides by velocity and then that gives you time is equal to displacement divided by velocity and so we can apply that to this right over here our displacement is 720 meters to the east so in this case our time is equal to 720 meters to the east 720 meters east is our displacement and we want to divide that by the given velocity will they give us the velocity of 3 meters per second to the east 3 meters per second to the east and once again 720 divided by 3 will give you 240 will give you 240 and then you have when you take meters in the numerator and you divide by meters per second the denominator that's the same thing as multiplying by seconds per meter those cancel out and you're just left with seconds here one note I want to give you in the last few problems I've been making vector quantities by saying to the east or going north and what you're going to see as we go into more complex problems and this is what you might see in typical physics classes or typical books is that you define a convention that maybe you'll say the positive direction especially when we're just dealing with when we're just dealing with one dimension whether you can either go forward or backwards or left or right we'll talk about other vector quantities when we can move in two or three dimensions but it might take some some kind mention like positive means you are moving to maybe you're moving to the east and maybe negative means you're moving to the west and so that way it'll it'll make wealth in the future we'll see it'll make the math a little bit the math will produce the results that we see a little bit better so if this would just be a positive 720 meters this would be a positive 3 meters per second and that implicitly tells us that that's the East if it was negative it would then be to the west something to think about we're going to start exploring that a little bit more in future videos and we might have maybe we might say positive is up negative is down or who knows there's different ways to define it when you're dealing in one direct one one dimension