Kinematic formulas and projectile motion
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Plotting projectile displacement, acceleration, and velocity
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, final velocity, and acceleration, all of those as functions 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 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 plus our acceleration times change in time. Right? If we start at some initial velocity, and then you multiply the acceleration times time. This part tells you how much faster or slower you're going to go than your initial velocity. And that will be, 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, 19.6 meters per second. So let's plot these out a little bit. Let's plot these out. So 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 I could call this the change in time axis. Actually, lets 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 5 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. So this is displacement. Displacement graph. And I want to, at the same time as 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. Time will only be positive. So once again, I care about 1 second, 2 seconds, 3 seconds, and 4 seconds in time. And velocity, I'm going to call this. This is going to be 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. This axis right here is time. So this is 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, and I'll just do this right from the get go, because we're going to assume that acceleration is constant. So this is 1 second, 2 seconds, 3 seconds, and 4 seconds into it. And then let's call this 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 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 table here. So in one column, I will do change in time. Or sometimes you could 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 what our displacement is. And I will do it for times 0, 1, 2, 3, 4. Or change in time. So when 0 seconds have gone by, when 1 second has gone by, when 2 seconds, 3 seconds, and 4 seconds have gone by. Actually let, me call this the change in time axis, because this 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-- I didn't label it here-- this is my acceleration graph. And I'm going off of the screen. All right. So let's fill these things out. So at time 0, what is our what is our velocity? Well, if we use this expression right here, time 0, or delta t is equal to 0. This expression right here, is going to be 0. And it's just going to be our initial velocity. And in the last video, we gave our initial velocity, is going to be as 19.6 meters per second. So it is going to be 19.6 meters per second. And let me plot that over here. At time 0, it is going to be 19.6 meters per second. What is our initial displacement at time zero, or change in time 0? So you look at this up here. Well, our delta t is 0, so this expression is going to be 0, and this expression is going to be 0. 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 happens after 1 second has gone by? What is now our velocity? Well, our initial velocity, right over here, is 19.6. 19.6 meters per second, that was a given. And our acceleration is 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 1, because delta t is 1. So we get 19.6 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 second. So 19.6 meters per second minus 9.8 meters per second-- one of these seconds goes away when you multiply it by second-- gives you 9.8 meters per second. So after 1 second, our velocity is now half of what it was before. So 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 of the information that we know. So we know that displacement is going to be equal to our initial velocity, which is 19.6-- and I won't write the units here, just for the sake of space-- times our change in time. I'll do it in that same color, so you what's what. Times our change in time plus 1/2. Now let me be clear, 1/2 times negative 9.8 meters per second squared. So 1/2 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 1/2, so this is going to be negative 4.9. All I did, is I took 1/2 times negative 9.8 over here. 1/2 times negative 9.8. And this is important. And this is why the vector quantities start to matter. Because if you didn't, if you put a positive here, you wouldn't have the object actually slowing down as it went up, because you would have gravity somehow accelerating it as it went up. But it's actually slowing it down. 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 meters per second squared 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-- I'll get my trusty TI-85 out. When one second has passed, the displacement is 19.6 times 1. Well, that's just 19.6. Minus 4.9 times 1 squared. So that's just minus 4.9. Gives us 14.7 meters. So after 1 second, the ball has traveled 14.7 meters in the air. So that's roughly over there. Now, what happens after 2 seconds? I'll do this in magenta. So after 2 seconds, our velocity is 19.6 minus 9.8 times 2. 2 seconds have gone by. Well, 9.8 meters per second squared times 2 seconds gives us 19.6 meters per second. So these just cancel out. So we get, our velocity is now 0. So after 2 seconds, our velocity is now 0. Actually, let me-- So let me make it so it's-- this thing should look more like a line. I don't make you get a sense-- So this is-- So let me just draw the line like this. Our velocity is now 0 after 2 seconds. What is our displacement? So we're literally at the point where the ball has no velocity. At exactly 2 seconds. So it's kind of gone up, and for that exact moment in time, it is stationary. And then what do we have going on in our displacement? We have 19.6-- let me get the calculator out for this. I could do it by hand, but for the sake of quickness-- 19.6 times 2-- 19.6 times 2 seconds minus 4.9 times 2 seconds squared. This is 2 seconds squared. Oh, I lost the calculator. Times 2 seconds squared. So that's times 4. So that gives us 19.6 meters. So we're at 19-- let me do that in magenta-- we are at 19.6 meters. So after 2 seconds, we are 19.6 meters in the air. Now let's go to 3 seconds. So after 3 seconds, our velocity is now-- I'll just get the-- it's 19.6 meters per second minus 9.8 times 3. And we could do that in our head, but just to verify it for us, let me get the calculator out. It's 19.6 minus 9.8 times 3. That gives us negative 9.8 meters per second. So after 3 seconds, our velocity is now negative 9.8 meters per second. What does that mean? It's now going in the downward direction at 9.8 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. At any time, I encourage you to pause it, and try it for yourself. So now, what is-- now this is a little, OK. So I'm looking at my displacement, I wrote right over here. So our displacement where delta t is 3 seconds, 19.6 times 3 minus 4.9 times-- and this is delta t, so this is 3 seconds. We're talking about when delta t, or our change in time, is 3 seconds. So that squared. So times 9. And that gives us 14.7 meters. So after 3 seconds, we're at 14.7 meters again. And so we're at the same position we were at 1 second, but the difference is, now we're moving downwards. Over here we were moving upwards. And then finally, what happens after 4 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 19.6 minus 9.8 times 4 seconds. Which is minus 19.6 meters per second. So 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. And what is our displacement? Get the calculator out. So we have, our displacement is 19.6 times 4-- 4 seconds have gone by-- minus 4.9 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. Actually, I can do a better job than that. I'll do a dotted line. Dotted lines are always easier to adjust midstream. If you plot displacement versus time, it looks something like this. Its 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 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 for this ball, or this rock-- and I know this video is getting long-- as it goes through the air, I'm going to draw the vectors for velocity. And I'm going to do that in orange, or no, maybe I'll do that in blue. So velocity in blue. So right when we start, it has a positive velocity of 19.6 meters per second. So I'll draw a big vector like this, 19.6 meters per second, that's its velocity. Then after 1 second, it's 9.8 meters per second. So it's half of that. Maybe it would look something like this, 9.8 meters per second. Then at this peak right over here, it has a velocity of 0. Then as you go to 3 seconds, the magnitude of its velocity is 9.8 meters per second. But it is now downwards. 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 19.6 meters per second. 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 was negative. It was negative 9.8 meters per second squared. And I'll do that in orange. So the acceleration, over here, negative-- no, 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.