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# Plotting projectile displacement, acceleration, and velocity

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

## Video transcript

what I want to do in this video now that we have displacement as a function of time given constant acceleration and an initial velocity I want to plot displacement velocity I should say final velocity and acceleration all of those as functions of time and just so that we really understand what's happening is the ball is going up and then down so we know this is our this is our this is our displacement as a function of time we know what our final velocity is going to be as a function of time we talked about it in the last video our final velocity is going to be our initial velocity our initial velocity plus our acceleration times change in time right if we start at some initial velocity the extend then you multiply the acceleration times time this is this part tells you how much faster or slower you are going to go then your initial velocity and that will be your I guess you could say your current velocity or the final velocity at that point in time and of course our acceleration we know our acceleration is pretty straightforward the acceleration due to gravity is just going to be negative 9.8 meters per second squared once again negative being the convention that it is in the downward direction our initial velocity is going to be in the upward direction nineteen point six meters per second so let's let's plot these out a little bit let's plot these out so the first this the first graph I want to do right over here will be my displacement versus time so this axis right over here is going to be time or maybe Becket's call this the change in time axis actually let's just call it time let's just call it time and then this axis right over here I will call displacement and let me put some markers here so let's say that this is five meters 10 meters 15 meters and 20 meters and then in the time this is 0 this is 1 this is 2 this is 3 and this is 4 seconds so this is in seconds right here this is meters 5 10 5 10 15:20 and then so this is displacement this is displacement displacement to graph and I want to at the same time is that I want to do a velocity graph so let me draw my velocity graph like this I'll do it a little bit different so this is because the velocity will be going up and down so we need to have positive and negative values here but time will only be positive and so once again I care about one second two seconds three seconds and four seconds in time and velocity I'm going to call this this is going to be ten meters per second 10 meters per second this is 20 meters per second this will be negative 10 meters per second and this will be negative 20 meters per second and so all of this is in meters per second this right here is velocity velocity this right here this axis right here is time so this is my velocity graph my velocity graph and why don't we just throw an acceleration graph over here although that's to some degree the easiest of them all so the acceleration graph the acceleration and I'll just do this right from the get-go because we're going to assume that acceleration is constant so this is one second two seconds three seconds and four seconds into it and then let's say let's call this negative 10 negative 10 and all of this is in meters per second squared and so we know our acceleration is negative 9.8 meters per second squared so the acceleration the acceleration the entire time over the four seconds the acceleration over the four seconds is going to be that's about negative 9.8 it's going to be that it's going to be a constant acceleration the entire time but let's figure out displacement and velocity so let me draw a little let me draw a little table here let me so I'll do in one column I will do change in time or you could sometimes you can do that as time let's figure out what our final velocity is or I should really say our current velocity or velocity at that time and then in this column I'll figure out our displacement is what our displacement is and I will do it I will do it four times zero one two three four or change in time so when zero seconds have gone by when one second has gone by when two seconds three seconds and four seconds have gone by actually let me call this the change in time axis because it is essentially how many seconds have gone by so this is my change in time axis and let me make it clear that this graph er I didn't label it here this is my acceleration graph acceleration and I'm going off of the screen all right so let's let's fill these things out so at time zero what is our what is our velocity well if we use if we use this expression right here at time zero or delta T is equal to zero this expression right here is going to be zero and it's just going to be our initial velocity and at the in the last video we gave our initial velocity is going to be as nineteen point six meters per second so it is going to be nineteen point six meters per second and let me plot that over here at time zero it is going to be nineteen point six meters per second what is our initial displacement at time zero our change in time zero so you look at this up here well our delta T is zero so this expression is going to be zero and this expression is going to be zero so we haven't done any displacement yet when no time has gone by so we have done no displacement so we have done no displacement we're right over there now what happens after one second one second has gone by what is now our velocity well our initial velocity right over here is nineteen point six nineteen point six meters per second that was a given and our acceleration is it's a negative 9.8 meters per second squared so it's a negative right over there and then you multiply that times delta T in every situation so in this situation we're going to multiply it by one because delta T is 1 so we get nineteen point six minus 9.8 that gives us exactly 9.8 meters per second and the units work out because you multiply this times seconds this gives you meters per sec so nineteen point six meters per second minus nine point eight meters per second one of these seconds goes away when you multiply it by second gives you 9.8 meters per second so after one second our velocity is now half of what it was before so we're now we're now going 9.8 meters per second let me draw a line here 9.8 meters per second now what is our displacement so you look up here and let me rewrite this displacement formula with all the information that we know so we know that displacement is going to be equal to our initial velocity which is nineteen point six and I won't write the unit's here just for the sake of space times our change in time times our doing that same color so you see what's what times our change in time plus one-half let me clear it one-half times negative 9.8 meters per second squared so one-half times a is going to be actually I can rewrite this right over here because this is going to be negative 9.8 meters per second times one-half so this is going to be negative 4.9 all I did is I took one-half times negative 9.8 over here one half times negative 9.8 this is important and this is why the vector quantity start to matter because if you didn't if you put a positive here you would you wouldn't have the object actually slowing down as it went up because you would have gravity somehow accelerating it is it went up but it's actually slowing it down it's pulling it's it's accelerating it in the downwards direction so that's why you have to have that negative right over there that was our convention at the beginning of the last video up is positive down is negative so let's focus so this part right over here negative 4.9 m/s squared times delta T squared times delta T times delta T squared and this will make it a little bit easier although we'll still let me get the calculator out so when one second has passed when one second is passing in my trusty ti-85 out when one second splats the displacement is nineteen point six times one well that's just nineteen point six minus 4.9 times one squared so that's just minus 4.9 minus 4.9 gives us fourteen point seven meters so fourteen point seven meters so after one second the ball has traveled fourteen point seven meters in the air fourteen point seven so that's roughly over there now what happens after two seconds I'll do this in magenta so after two seconds our velocity is nineteen point six minus nine point eight times two times two this is two seconds have gone by well nine point eight times to get nine point eight meters per second squared times two seconds gives us nineteen point six meters per second so these just cancel out so we get our velocity is now zero so after two seconds our velocity is now zero actually let me so let me make it so it's this this thing should be this thing should look more like a line I don't want to make you get a sense so this is so let me just draw the line like this so our velocity is now zero after two seconds what is our displacement our displacement so we're literally at the point where the ball has no velocity at exactly two seconds so it's kind of gone up and it's right for what important for just that that exact moment in time it is stationary and then what do we have going on in our displacement we have nineteen point six let me get the calculator out for this we can do it by hand but for spit sake of of quickness nineteen point six times to nineteen point six times two seconds minus four point nine times two seconds squared this is two seconds squared I lost the calculator times two seconds squared so that's times four so that gives us nineteen point six meters so we are at nineteen let me defend a magenta we are at nineteen point six nineteen point six meters so after two seconds we are nineteen point six meters in the air now let's go to three seconds so after three seconds well our our velocity is now I'll just get B it's nineteen point six meters per second minus nine point eight times three and we could do that in our head but just to verify it for us let me get the calculator out it's nineteen point six minus nine point eight times three that gives us negative nine point eight meters per second negative nine point eight meters per second so after three seconds our velocity is now negative nine point eight meters per second what does that mean it's now going in the downward direction at nine point eight meters per second so this is our velocity graph and then what is our displacement at this point so once again let's get the calculator out if you're getting the hang of this and any time I encourage you to pause it and try it for yourself so now what is this is a little okay so I'm looking at my displacement I wrote it right over here so our displacement where delta T is three seconds nineteen point six times three minus four point nine times and this is delta T so this is three seconds we're talking about when delta T or change in time is three seconds so that's squared so times nine and that gives us fourteen point seven meters so it fourteen point seven meters so after three seconds we're at fourteen point seven meters again and so we're the same position we're at one second but the difference is is now we're moving downwards over here we were moving upwards and then finally what happens after four seconds well what's our velocity well let me just get the calculator out although you might be able to figure this out in your head our velocity is going to be nineteen point six minus nine point eight minus nine point eight times times four seconds times 4 which is minus nineteen point six meters per second minus nineteen point six meters per second so we're going our magnitude of our velocity is the same as when we initially threw the ball except now it's going in the opposite direction it's now going downwards so it's now going downwards and what is our displacement get the calculator out well so we have our displacement is nineteen point six times four four seconds have gone by minus four point nine times 4 squared which is 16 so times 16 which is equal to 0 our displacement is 0 we are back on the ground we are back on the ground so if you were to plot its displacement you would actually get a parabola a downward-opening parabola that looks something like this doing my best to draw it relatively neatly so I check it to a better job than that I'll do a dotted line dotted lines are always easier to adjust midstream so the if you draw it if you plot this displacement versus time it looks something like this it's velocity is just this downward sloping line and then the acceleration is constant and the whole reason why I wanted to do this is I wanted to show you that the velocity the whole time is decreasing at a constant pace and that makes sense because the rate at which the velocity increases or decreases is the acceleration and the acceleration based are based on our convention is downwards so that's why it's decreasing we have a negative slope here we have a negative slope of negative 9.8 meters per second squared and so just to think about what's happening in this ball for this ball or this rocket I know this video is getting long as it goes through the air I'm going to draw the vectors for velocity so I'm going to do that in orange or maybe I'll do that in blue so velocity in blue so right when we start it has a positive velocity of nineteen point six meters per second so I'll draw a big vector like this 19 point six meters per second that's its velocity then after one second it's 9.8 meters per second so it's half of that so then it's maybe it would look something like this 9.8 meters per second then at this peak right over here it has a velocity of zero then as you go to three seconds it has nine point it has the magnitude of its velocity is 9.8 meters per second but it is now downwards it is now down where it so it looks like this and then finally right when it's when it hits the ground it has right before it hits the ground it has a negative velocity of nineteen point six meters per second so it would look like it would look like this roughly like this if I use the same scale over here but what was the acceleration the entire time well the entire time the acceleration was negative it was negative 9.8 meters per second squared and I'll do that in orange so the acceleration over here negative now I want to do that in orange the acceleration was negative 9.8 meters per second squared acceleration negative 9.8 meters per second squared negative 9.8 meters per second squared the acceleration is constant the entire time this last one is negative 9.8 meters per second squared it does not change depending where you are in the curve when you're near the surface of the earth so hopefully that clarifies things a little bit and gives you a good sense of what happens when you throw a projectile into the air