Anderson Video - Projectile Motion Example: Time in Air

Professor Anderson
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>> Hello, class. Professor Anderson here. Let's revisit the projectile motion problem that we looked at a second ago, which was let's throw an object off of a building with an initial speed Vi but upwards at an angle theta, and let's see if we can solve for how long this thing is in the air, okay, and we gave you some givens. We said that Vi was 30 meters per second, theta was-oh, we said 20, didn't we? 20 meters per second for Vi? 20 meters per second. Theta was 30, and let's say that you're sitting up there at a height 45 meters. Okay, what do we do? We want to solve for the time that it's in the air. What should I do next? Somebody give me a thought. Yeah? >> Go to our kinematic equations? >> Go to our kinematic equations. Which one do you like? >> I have no idea. >> Okay, what variables should be used, x or y? >> Probably y. >> Probably y. Okay, so let's start it out. Y final equals-What goes over here? Y initial plus what else? Anybody remember this stuff? Yeah? >> Vy initial. >> Vy initial times t. >> Plus 1/2 a t squared. >> Good. Burned into your brain. Burn these into your brain. Do it now, okay? We're already burning it into the screen, but burn it into your brain as well. Okay. That looks good, and we know a lot of this stuff, right? What's Y final when it hits the ground? Well, that's zero. We can make y equals zero wherever we want, but ground level's a natural choice. Y initial is where we started from, height h. Vy initial is what? Well, we want the vertical component of the velocity, which means Vi sine theta. Okay, that's the vertical component of our initial velocity, and then we know that a y is negative g, and so look at this equation. This is a quadratic equation. We can rewrite it in a form that you are maybe happier with, which is 1/2 g t squared minus Vi sine theta times t minus h equals zero, and just for kicks, let's multiply everything by 2. So this will become g t squared minus 2 Vi sine theta times t minus 2 h equals 0, and now that's a quadratic equation, and we can solve it for t. So remember what a quadratic equation looks like. A quadratic equation usually has the form (a x squared + b x + c) = 0, and you probably remember what the solution is, the quadratic formula. It is (negative b plus/minus the square root (b squared - 4 a c)) all over 2 a. So in our case, this is a, this is b, and this is c. So we can write t equals negative b. b is a negative number, so a negative negative becomes positive, and we get 2 Vi sine theta plus/minus the square root of b squared, (2 Vi sine theta) squared, minus 4 a c, of b squared, (2 Vi sine theta) squared, minus 4 a c, 4 times a is g; c is negative 2 h, and now all of this is over 2 a, which is 2 g. All right, and we can just rewrite this again. 2 Vi sine theta plus/minus the square root of (4 Vi squared sine squared theta plus 8 g h) and all of that over 2 g, and now this is where I want you to grab your calculators and plug in some of these numbers. So it's 2 times 20 times sine of 30 degrees plus/minus the square root of (4 (20 squared) (sine of 30 degrees squared) plus 8 times 9.8 times 45, and then all of this over 2 times 9.8. So why doesn't somebody do the square root first and tell me what you get for the square root. >> It's 62.67. >> The square root is sixty what? >> 62.67. >> 62.67. Anybody else concur on that one? Okay, you concur? All right, excellent. So let's do the other stuff. What is this first part? 40 times sine 30. What do you guys get for that? >> 20. >> 20? Okay, that makes sense, right? 30, 60, 90 (1 squared) 3 2. So sine of 30 is 1/2. Good. So we have 20 plus/minus 62.67, and then we're dividing the whole thing by 19.6. So what do we get for our two possibilities? >> 4.21. >> 4.21? >> Yes. >> Positive? >> Yes. >> 4.21 is one of the answers, and what's the other one? >> Negative 2.1. >> Negative 2.1, okay, and both of these are seconds. Okay, so you look at those, and you say, "All right, those are my two answers." Which one do you like? How long is it in the air? What are you going to punch in for your homework? >> 4.21. >> Okay, 4.21 seconds. That's how long it's in the air. What about the negative part? All right, what does that mean? Well, we have this plus/minus on our square root. We get a negative solution as well. Let's think for a second about this negative solution and see if it has any physical significance, right? We just jumped at the 4.21 answer because we like things that are positive, right? When you talk about how long, it's a positive number. What's the time in seconds? But what about this negative number? All right, the math just fell out and gave us this negative 2.1 seconds. What does that mean? Well, let's go back to our picture for a second. Here is our person. They are going to launch this object, and it's going to follow a parabola, okay. So starting from there, going to there, it's going to take 4.21 seconds. What do you think the negative 2.1 seconds means? Anybody have an idea about that? Yeah. >> Wouldn't that be when the ball or the object passed, goes below you, like the time it takes from when it passes you to hit the ground would be 2.1 seconds. >> Okay. I'm not exactly sure what you mean by that, but you're saying when it passes us to hit the ground, meaning from here to the bottom? Or something else? >> So it curves up and down like that. So once it goes up and you see it again, that would be like 4.21 minus 2.1. That's how long it took. >> Okay. >> From there to down will be 2.1. >> Okay, I think you are exactly right, but I'm going to say it slightly differently, which is the following. When we said it's 4.21 seconds from there to there, we said let's start our clock right when it's at my hand, but let's say our buddy was standing down here, and they toss this object from there, and it flew past me, and when I started my clock when it flew past me, it took 4.21 seconds to hit the ground. If I went back in time and said, "How much earlier did my friend toss the ball from here," it would be negative 2.1 seconds, okay. That's what that means. If I reverse the motion, it would take 2.1 seconds going the other way to get back down to the ground. Kind of cool, right? And I think that's basically what you were trying to say in terms of these horizontal positions, right. When it came back down to me at that point, then it would be 2.1 seconds from there on. Very good. Okay, questions about that problem? Okay, hopefully that's clear, reasonably clear. If not, definitely come see me in office hours. Cheers.
>> Hello, class. Professor Anderson here. Let's revisit the projectile motion problem that we looked at a second ago, which was let's throw an object off of a building with an initial speed Vi but upwards at an angle theta, and let's see if we can solve for how long this thing is in the air, okay, and we gave you some givens. We said that Vi was 30 meters per second, theta was-oh, we said 20, didn't we? 20 meters per second for Vi? 20 meters per second. Theta was 30, and let's say that you're sitting up there at a height 45 meters. Okay, what do we do? We want to solve for the time that it's in the air. What should I do next? Somebody give me a thought. Yeah? >> Go to our kinematic equations? >> Go to our kinematic equations. Which one do you like? >> I have no idea. >> Okay, what variables should be used, x or y? >> Probably y. >> Probably y. Okay, so let's start it out. Y final equals-What goes over here? Y initial plus what else? Anybody remember this stuff? Yeah? >> Vy initial. >> Vy initial times t. >> Plus 1/2 a t squared. >> Good. Burned into your brain. Burn these into your brain. Do it now, okay? We're already burning it into the screen, but burn it into your brain as well. Okay. That looks good, and we know a lot of this stuff, right? What's Y final when it hits the ground? Well, that's zero. We can make y equals zero wherever we want, but ground level's a natural choice. Y initial is where we started from, height h. Vy initial is what? Well, we want the vertical component of the velocity, which means Vi sine theta. Okay, that's the vertical component of our initial velocity, and then we know that a y is negative g, and so look at this equation. This is a quadratic equation. We can rewrite it in a form that you are maybe happier with, which is 1/2 g t squared minus Vi sine theta times t minus h equals zero, and just for kicks, let's multiply everything by 2. So this will become g t squared minus 2 Vi sine theta times t minus 2 h equals 0, and now that's a quadratic equation, and we can solve it for t. So remember what a quadratic equation looks like. A quadratic equation usually has the form (a x squared + b x + c) = 0, and you probably remember what the solution is, the quadratic formula. It is (negative b plus/minus the square root (b squared - 4 a c)) all over 2 a. So in our case, this is a, this is b, and this is c. So we can write t equals negative b. b is a negative number, so a negative negative becomes positive, and we get 2 Vi sine theta plus/minus the square root of b squared, (2 Vi sine theta) squared, minus 4 a c, of b squared, (2 Vi sine theta) squared, minus 4 a c, 4 times a is g; c is negative 2 h, and now all of this is over 2 a, which is 2 g. All right, and we can just rewrite this again. 2 Vi sine theta plus/minus the square root of (4 Vi squared sine squared theta plus 8 g h) and all of that over 2 g, and now this is where I want you to grab your calculators and plug in some of these numbers. So it's 2 times 20 times sine of 30 degrees plus/minus the square root of (4 (20 squared) (sine of 30 degrees squared) plus 8 times 9.8 times 45, and then all of this over 2 times 9.8. So why doesn't somebody do the square root first and tell me what you get for the square root. >> It's 62.67. >> The square root is sixty what? >> 62.67. >> 62.67. Anybody else concur on that one? Okay, you concur? All right, excellent. So let's do the other stuff. What is this first part? 40 times sine 30. What do you guys get for that? >> 20. >> 20? Okay, that makes sense, right? 30, 60, 90 (1 squared) 3 2. So sine of 30 is 1/2. Good. So we have 20 plus/minus 62.67, and then we're dividing the whole thing by 19.6. So what do we get for our two possibilities? >> 4.21. >> 4.21? >> Yes. >> Positive? >> Yes. >> 4.21 is one of the answers, and what's the other one? >> Negative 2.1. >> Negative 2.1, okay, and both of these are seconds. Okay, so you look at those, and you say, "All right, those are my two answers." Which one do you like? How long is it in the air? What are you going to punch in for your homework? >> 4.21. >> Okay, 4.21 seconds. That's how long it's in the air. What about the negative part? All right, what does that mean? Well, we have this plus/minus on our square root. We get a negative solution as well. Let's think for a second about this negative solution and see if it has any physical significance, right? We just jumped at the 4.21 answer because we like things that are positive, right? When you talk about how long, it's a positive number. What's the time in seconds? But what about this negative number? All right, the math just fell out and gave us this negative 2.1 seconds. What does that mean? Well, let's go back to our picture for a second. Here is our person. They are going to launch this object, and it's going to follow a parabola, okay. So starting from there, going to there, it's going to take 4.21 seconds. What do you think the negative 2.1 seconds means? Anybody have an idea about that? Yeah. >> Wouldn't that be when the ball or the object passed, goes below you, like the time it takes from when it passes you to hit the ground would be 2.1 seconds. >> Okay. I'm not exactly sure what you mean by that, but you're saying when it passes us to hit the ground, meaning from here to the bottom? Or something else? >> So it curves up and down like that. So once it goes up and you see it again, that would be like 4.21 minus 2.1. That's how long it took. >> Okay. >> From there to down will be 2.1. >> Okay, I think you are exactly right, but I'm going to say it slightly differently, which is the following. When we said it's 4.21 seconds from there to there, we said let's start our clock right when it's at my hand, but let's say our buddy was standing down here, and they toss this object from there, and it flew past me, and when I started my clock when it flew past me, it took 4.21 seconds to hit the ground. If I went back in time and said, "How much earlier did my friend toss the ball from here," it would be negative 2.1 seconds, okay. That's what that means. If I reverse the motion, it would take 2.1 seconds going the other way to get back down to the ground. Kind of cool, right? And I think that's basically what you were trying to say in terms of these horizontal positions, right. When it came back down to me at that point, then it would be 2.1 seconds from there on. Very good. Okay, questions about that problem? Okay, hopefully that's clear, reasonably clear. If not, definitely come see me in office hours. Cheers.