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Plotting projectile displacement, acceleration, and velocity

Video transcript
What do I do in this video now that we have displacement as a function of time given constant acceleration and initial velocity I want to plot displacement velocity, final velocity, and acceleration all those function of time, just so we really understand what's happening as the ball is going up and then down So we know this is our displacement function of time, we know what our final veloctiy is going to be, as a function of time we talked about in our last video the final velocity is going to be the initial velocity plus our acceleration times change in time right if we start we some initial velocity, and you multiply the acceleration with time this part tell you how much faster or slower you are going to go between initial velocity or that will be your I guess current velocity the final velocity at that point in time and of course the acceleration we know our acceleration is pretty straigth forward the acceleration due to gravity is just gonna be negative 9.8 m/s*s once again the negative being the convention that it is in the downward direction our initial velocity is going to be upper direction 19.6m/s so let's plot this out a little bit, let's plot these out so the first graph I wanna do will be my displacement verses time so this axis right over here is going to be time and let me just call it change in time, actually I'll just call it time let's just call it time and then this axis right over here I will call it displacement and we leave some marker here so let's say that this is 5m, 10m, 15m and 20m and the time, this is zero, this is 1 this is 2 this is 3 & this is 4 seconds, so this is in second here this is meters 5,10,15,20 and then so this is displacement displacement graph, and I wanna do at the same time a velocity graph let me draw my velocity graph like this, a little bit different so this is the velocity will be going up and down, so we have positive and negative values here, time will only be positive so once again I care about 1s 2s 3s and 4s and velocity I'm gonna call this, this is going to be 10m/s 10m/s this is 20m/s, this will be negative 10m/s and this will be negative 20m/s, all of these are in m/s this right here is velocity, this axis right here is time so this is my velocity graph, and why don't we draw an acceleration graph over here, to some degree it's easiest of them all so the acceleration graph, so I will just do the fraction here and assume that the acceleration is a constant, so this is 1s 2s 3s and 4s and let's call this negative 10 and all of these are meters per s square so we know our acceleration is 9.8 m/s*s so the acceleration the entire time over the four seconds the acceleration over the four seconds is going to be negative 9.8 it's gonna be a constant acceleration the entire time let's figure out the displacment and velocity, let me draw a table here, in one column I will do change in time and you can sometime do that as time let's figure out what our final velocity is I should say the current velocity of velocity at that time and in this column I will figure out what displacement is and I will do it in time 0 1 2 3 4 so in 0 zero second gone by when 1 second has gone by, when 2s 3s 4s has gone by Actually let me call it change in time axis this is essentially how many seconds it has gone by So this is my change in time, let me make it clear that this graph here is acceleration graph acceleration I will put it on the screen Alright, so let's fill these things out so times zero, what is our velocity? well if we use this expression right here time zero or delta t equal zero this expression right here is gonna be zero, and it's just going to be initial velocity, in the last video we gave our initial velocity is going to be 19.6m/s, so it is going to be 19.6m/s I will plot that over here, time zero, it's going to be 19.6 m/s what is our initial displacement at time zero? Our change in time zero, so you look at this up here delta t is zero so this expression here 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, we are right over there now what happen after 1s? 1s has gone by what is now our velocity? well our initial velocity right here is 19.6m/s that was a given, and our acceleration is negative 9.8m/s*s so it's negative right over there and you multiply that by delat t in every situation, so in this situation we are gonna multiply it by 1 delta t is 1, so you have 19.6 minus 9.8 that gives exactly 9.8m/s and the unit we got cause we multiply here is second so its give us meters per second 19.6m/s minus 9.8m/s one of these seconds go away multiply by second give you 9.8m/s so after 1s our velocity is now half of what it was before so we are now going 9.8m/s, let me draw a line here 9.8m/s, now what is our displacement? so you look up here Let me rewrite this displacement formular here with all the infomation So we know that displacement is going to be equal to initial veloctiy which is 19.6m/s now I won't write the units here just for the sake of space times change in time, times our, use the same color to see what is what times our change in time, plus one half let me be clear one half times negative 9.8m/s*s so one half times a is going to be I rewrite this right over here cause this is gonna be negative 9.8m/s*s times one half, so this is going to be negative 4.9 All I did is one half times negative 9.8 over here It is important that is why the vector quantity is start to matter because if you put a positive here you wouldn't have the obeject slowing down as you went up, because you will have gravity somehow accelerating as it went up, but it's actually slowing it down it's pulling it, it's accelerating it in downward direction so that's why you have a negative right over there, that was out convention at the beginning of last video, up is positive, down is negative, so let's focus So this part right over here, negative 4.9 m/s*s times delat t square times delta t, times delta t square, this will make it a little bit easier Although it still, let me get the calculator out So when one second has past, I let my trusty TI-85 out now when 1s past it's 19.6 times 1, well that's just 19.6 minus 4.9 times 1 square So that's just minus 4.9, mius 4.9, gives us 14.7 meters So 14.7 meters So after 1s, the ball has travel 14.7 meters in the air So that's roughly over there, now what happen after 2s? I'll do the same agenda, so after 2s, our velocity is 19.6 minus 9.8 times 2, times 2 this is 2s has gone by well 9.8 times 2, 9.8 times 2 square seconds gives us 19.6 m/s so these just cancel out, so we get out velocity now is zero So after two seconds our velocity now is zero let me make it so this thing should more look like a line I don't get a sense, so this is let me just draw the line like this so our velocity now is zero after 2 seconds what is our displacement? So literally we at the point with the ball has no velocity exactly two seconds, it kind of gone up, and right for that exact moment of time it's stationary and then what do we have going on in our displacement? We have 19.6 let me get the calculator out for this We can do it by hand but for the sake of quickness 19.6 times 2 minus 4.9 times 2s squared, this is 2s squared so that's times four, so that gives us 19.6 meters So we have 19, we are at 19.6 meters after 2 seconds we are 19.6 meters in the air now let's go to 3s, so after 3s, our velocity now it's 19.6 m/s minus 9.8 times 3, we could do that in our head but just verified it for us, let me get the calculator out it's 19.6-9.8*3, that gives us 9.8 m/s, negative 9.8m/s So after 3s, our velocity is now negative 9.8 m/s, what is that mean? it's now going in downward direction, 9.8 m/s, so this is our velocity graph, and what is our displacement? So once again, let's get the calculator out you are getting the hang of this at anytime I encourage you to pause it and try it for yourself So now what is this is, Okay I'm looking at the displacement Our displacment was delta t with 3s 19.6*3 - 4.9 times so this is delat t so this is 3s we are talking about delat t, our change in time is 3s so that's square So times 9 and that gives us 14.7 meters, so 14.7 meters So after 3s we are 14.7m again, the same position with 1s but the difference is now we are moving downard over here we were moving upwards and finally what happen after 4s? Or what's our velocity? Let me just get the calculator out or you might be figure this out in your head Our velocity is going to be 19.6 - 9.8 times 4s just minus 19.6 m/s So we are going a magnitude of velocity which is the same initially threw the ball except now it's going at opposite direction it's now going downward, so it's now going downward and what is our displacement? get the calculator out So we have out displacement is 19.6 times 4, 4s has gone by minus 4.9 times 4 square which is 16 times, which is equal to zero! the displacement here is zero! We are back on the ground So if you plot the displacement, you will actually got a parabola a downward opening parabola that looks something like this I best draw it relatively neatly So my check to do it better than that Dotted line, dotted line is always easier to adjust than its streamed so if you plot displacement verses time it look something like this it's velocity just downward sloping line, and the acceleration is constant and the whole reason why I wanted to this is I wanted to show you that velocity the whole time is decreasing at constant rate and that make sense because that's the rate which the velocity increases and decreases as the acceleration, and the acceleration base on our convention is downward so that why it's decreasing, we have a negative slope here we have a negative slope of negative 9.8 m/s*s and just to think about what's happening for this ball I know this video is getting long as it goes I'm gonna draw the vectors for velocity So I'm gonna do that in orange, or maybe I will do that in blue So velocity in blue, so right when we start, it has a positive velocity of 19.6m/s, so I will draw a big vector like this 19.6m/s that's velocity, but after 1s is 9.8m/s so half of that so then its maybe would look something like this 9.8m/s, then at the speed right over here has the velocity of zero then as you go to 3s, the magnitude of the velocity is 9.8m/s but it is now downwards, so it looks like this and then finally right when it hits the ground, right before it hits the ground it has a negative velocity of 19.6m/s So 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 is negative It's 9.8m/s*s I will do that in orange So the acceleration over here Negative, now I wanna do that in orange The acceleration is negative 9.8m/s*s The acceleration is constant the entire time So last one is negative 9.8m/s*s it does not change depending on where you are in the curve when you are near the surface of the earth So hopefully that clarify the things a little bit and gives you what happen when you throw up a projectile into the air