Two-dimensional projectile motion
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Total displacement for projectile
- [Instructor] Let's do another example where we are projecting something and it lands at a different level and we'll also figure out some other interesting things. We'll figure out what the actual velocity vector is when it's landing, so both its magnitude and its direction. So let's do a situation. So let's say we are launching something from ground level, and we're gonna launch it at a pretty steep angle over here. So let's say we launch it with an angle, let's say we launch it with an angle of, let me just, let's say it's 80 degrees. We launch it at an 80 degree angle, at, let me. We launch it at an 80 degree angle, 80 degree angle, and it is going to be going at 30 meters per second, 30 meters per second, so that's the length of this vector, that's the magnitude of that vector, and let's say we want to make it land on this landing, and this landing right over here, this landing right over here has a height, this landing over here has a height of 10 meters. This height over here is 10 meters, so what I wanna do first of all is I wanna figure out how far along the landing, how far along the landing do I actually land, how far along the landing. So, and maybe I'll add some other information right here. So say from this launching point to the beginning of the landing, let's say that this right over here is two meters, two meters, so we just want to know how far along the landing do we land. So like we did before, we want to break this vector into its horizontal and vertical components, its vertical and vertical components, and I'm gonna go a little bit faster in this video, 'cause hopefully we're getting the knack for this type of thing. So our velocity, our velocity in the vertical or our vertical component of our velocity is going to be equal to the magnitude of our total velocity, 30 meters per second, times, and it's going to be the sine of 80 degrees, 'cause sine is opposite over hypotenuse, times the sine of 80 degrees, and our, and we'll just get it out of the way right now, the horizontal component of our velocity is going to be 30 meters per second, and I'm not writing the units here, just to save space, times the cosine of 80 degrees. Once again, cosine is adjacent, adjacent, adjacent over hypotenuse, and if you feel like I'm skipping steps, in the last few videos, I go into this in much more detail. So, how much time do we spend in the air? How much time do we spend in the air? So, once again, in the last few videos, we saw that we can look at our, we can look at displacement, if we want to figure out time in the air, we know that displacement is equal to the initial velocity, initial velocity times time, plus acceleration, let me write times change in times, since that's more technically, that's technically more correct, plus acceleration times change in time squared, over, over two. Now, in our situation, we know what our initial velocity is, we are talking about the vertical direction right over here, so our initial velocity is going to be this. We are trying to figure out how time in the air, and the vertical component determines that, because at some point, when it hits back to the ground, it's not gonna be traveling anymore, so that's what's determining its time in the air. So we know that, we know the acceleration. Remember, our convention, when we are dealing with the vertical dimension is up is positive, down is negative, so this is negative 9.8 meters per second squared, and then we can, and then what is the total displacement that we are gonna have? Well, we are starting at ground level, and we are just talking about the vertical, remember that. So our total displacement is going to be 10 meters, so this value right here is going to be 10 meters, so it simplifies to 10 meters, I won't write the units here, is equals 10 is equal to, what's 30 sine of 80 degrees? So we have 30 times the sine of 80 degrees, gives us 29.54. 29.54. So this is 29.54 times our change in time, times our change in time, and so this is negative 9.8 divided by two, so it's negative, I'll do that same green color, four point, that's not the same green color, negative 4.9 meters per second squared. I'm not writing the units here, times delta t squared, times delta t squared, and then we can subtract 10 from both sides and write this in a traditional quadratic equation form, so we get negative, and I'm gonna flip the sides too, negative 4.9 times delta t squared, and then we have, plus 29.54 times delta t, minus 10, minus 10 is equal to zero, and then we can use the quadratic formula to find the roots of this, so the delta t's that satisfy this quadratic equation are going to be negative b, so negative 29.54, plus or minus the square root of 29.54 squared, b squared, minus four, minus four, times a, which is negative 4.9, for the negative times a negative is a positive. So it's plus four times positive 4.9, times, or I shouldn't have, I shouldn't have jumped so fast to get rid of the negative. So it's gonna be minus four times a, which is negative 4.9, times c, which is negative 10, so just a times c, negative 4.9 times negative 10, let me just write negative 4.9 times negative 10, these two guys, their signs are going to cancel out. All of that, all of that over, all of that over two a. All of that over negative 4.9 times two, so negative 9.8. And like we saw in the last video, we want a positive value for this. A negative time is nonsensical, that's kind of going into the past, so we want a positive value. And since we have a negative in the denominator, we want to have a negative value up here, and if we already have a negative value here, if we have subtract from that native value, we're definitely gonna have a negative value up here, and then you divide by a negative value, we'll get a positive value, so we can really focus on the subtracting the radical, and you could try it out. If you tried the positive version, you're gonna get a negative value for this entire thing. Do that on, you can try that out after this video, just to verify that you get a nonsensical answer. So let's use a negative right over here, so we have negative 29.54 minus the square root of, the square root of 29.54 squared, and then we have minus four, minus four, times negative 4.9 times negative 10, these two, when you take the product, it's positive 49, so times, times 49, and I should add a parentheses over there, so let me insert a parentheses. You can insert parentheses, okay, so times 49. So this right here will give me the numerator if I evaluate this, got me negative value, and I divide that by negative 9.8, so divided by negative 9.8 gives me 5.67, 5.67 seconds. This is equal to 5.67 seconds, and you could keep the units in there and make sure that all of the dimensional analysis works, and I think you'll find that it does. So our total, our total vertical, the total time in the air is 5.6, the total time in the air is 5.67 seconds. Now, what I want to do, the whole point of this is to figure out, is to figure out how far along this landing we land. Well, our horizontal component of our velocity is right over here. We know that, we know that our displacement in the horizontal direction, our displacement in the horizontal direction will be our velocity in the horizontal direction, and it's a constant velocity, so it's the same thing as our average velocity in the horizontal direction, times the change in time. Well, the change in time is, so I'll just that times, our change in time. So this is going to be equal to 30 cosine of 80 degrees, 30 cosine of 80 degrees, times 5.67 seconds, times 5.67 seconds, and this, I won't write the units. This is meters per second times seconds, it'll give us an answer in meters. So once again, we have, this is our time, this is the 5.67, times 30 cosine of 80 degrees, gives us 29.53 meters. So our total horizontal traveling, displacement I guess, we could say, is 29, I already forgot the number, 29.53 meters. 29.53 meters, and I'll write this as a vector, that is our horizontal displacement, it's equal to 29.53 meters. Now, we've done a lot of deconstructing vectors, what I thought would be interesting in this video is to construct a vector. So we know our horizontal displacement. We also know our vertical displacement, it's positive 10 meters, so what's our total displacement? Let me write this down. So we have a horizontal displacement. We have a horizontal displacement of 29.53 meters, and we have a vertical displacement of plus 10 meters. We have a vertical displacement of positive 10 meters, and so what is our total displacement? What is our total displacement going to be? Well, we could use the Pythagorean theorem now here. The square of the magnitude of our total displacement is going to be equal to the sum of these two squares, or another way, this is just the Pythagorean theorem, so we could have, if we call this length over here c, or let me just write it, this is the magnitude of our displacement right over here, the magnitude of our total displacement squared is going to be equal to 10 squared plus 29.53 squared. So it's going to be 10 squared plus 29.53 squared. I'm gonna do that same color, plus 29.53 squared. If we want to solve for this, we just take the square root of both sides, so I'll just do that in place. So if we just take the square root of both sides, we will get the magnitude of our total displacement, and I can get the calculator out once again to do that. So the magnitude of our total displacement is the square root of, 10 squared is just 100, plus 29, I can even use, well, I could use all of this information, so that I don't even lose some precision. I could say second answer, that literally means the previous answer, which is that 29.53 squared, gives us a total displacement of 31.18 meters. So this is equal to 31.18 meters, and of course, it's a vector, so we've only given you the magnitude, we also need the direction. So one way to specify direction is to give you the angle with the horizontal, and let's call that angle theta, and once again, we can use our trig, our trig functions over here. We could use pretty much any of the trig functions, but we know the opposite side is 10, we know the hypotenuse here is 31.18, so why not use sine, sine is opposite over hypotenuse. So we know that the sine of theta is going to be equal to 10 over 31.18. 10 over 31.18, or if you want to solve for theta, you take the arcsine of both sides, or the inverse sine of both sides, you get theta is equal to the inverse sine or I could write arcsine over here, of 10, of 10 over 31.8. 10 over 31.18, I should say. So let's once again get the calculator out to figure out that value, so I'm gonna take the inverse sine. Once again, so this is the same thing as the arcsine. This just says, give me the angle that when I take its sine, I get this value right over here, so the inverse sine of 10 divided by our previous answer, 31.18. I'll just say that our previous answer is equal to, so this says, give me the angle whose sine is 10 over 31.18. So I have 18.7 degrees, or 18.71 degrees, so this is equal to 18.71 degrees, 18.71 degrees above the horizontal. So here we've constructed a vector, we took its vertical component and its horizontal component and we were able to figure out the total vector, so this projectile in this situation is, its total displacement, so just to make it clear, its path will look something like this, its path will look something like this. Its path is gonna look something like this, and we've just calculated its total displacement, its total displacement is 31.18 meters, 18.71 degrees above the horizontal. And the other thing I've realized is when I started this problem, I asked you, I think I was asking you how far along the platform, and we've figured out its total horizontal displacement, so if you wanna know how far along the platform, the platform starts two meters to the right, so it's really 27.53 meters along the platform is where it lands.