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A stone is thrown from the tOp of building upward at an angle of 0, = 375 with thie horizontal the initial speed is 20.0 m/s as shown in the Figure. If the buil...

Question

A stone is thrown from the tOp of building upward at an angle of 0, = 375 with thie horizontal the initial speed is 20.0 m/s as shown in the Figure. If the building is 14.1 m height. Find the X-component of the velocity when the stone is above Sm from the ground?mls 18.1mls 34.1mls 24.1mls 16

A stone is thrown from the tOp of building upward at an angle of 0, = 375 with thie horizontal the initial speed is 20.0 m/s as shown in the Figure. If the building is 14.1 m height. Find the X-component of the velocity when the stone is above Sm from the ground? mls 18.1 mls 34.1 mls 24.1 mls 16



Answers

In Fig. $4-34$ , a stone is projected at a cliff of height $h$ with an initial speed of 42.0 $\mathrm{m} / \mathrm{s}$ directed
at angle $\theta_{0}=60.0^{\circ}$ above the horizontal. The stone strikes at $A$
5.50 $\mathrm{s}$ after launching. Find (a) the height $h$ of the cliff, (b) the speed of the stone just before impact at $A,$ and (c) the
height $H$ reached above the ground.

Here we have a scenario from the edge of the rooftop of a building, a boy stores a stone at an angle of 25 degrees above the horizontal, the stone hits the ground. 4.2 seconds later, it travels a distance of 105 m away from the base of the building. We're going to ignore air resistance in this situation. So it's right then what we know so far this angle but it was thrown as 25 degrees and call that 3 to 6 data is 25 degrees. The angle above the horizontal. Let it's thrown. The stone hits the ground at a change in time. Delta T. 4.20 seconds later. In this distance, it travels. The change in exposition is 105 m from the base of the building. So now we ask ourselves, what is the initial velocity of the stone? Just go ahead and draw it up. I'm gonna draw the initial velocity and we know that there's going to be some X. Component associated and some Y component associated. Of course this angle is 25 degrees. In order to find this X. Component. We would say that it is equal to this Visa X is equal to V times the Co sign 25 degrees right? Because we want that adjacent side. But we also know the change in X and the change in time. So that Visa bags can also be written as delta X over delta T. We have both of those values. Why can't I do this? Because gravity only acts in the Y direction in this case because it's project to emotion. So because the X velocity doesn't change, it's going to be a constant. So the average velocity will be equal to the velocity at any instantaneous time. So we can say that the is equal or sorry, the times of co sign 25 degrees is equal to the changing acts, which is 105 over the change in time, which is 4.20 seconds meters per second. So now all we have to do to get V. By itself is to 105 m over 4.20 seconds times the co sign of 25 degrees. Let's go ahead and plug that into a calculator, 105 m over 4.2 seconds times to co sign of 25 degrees. And we find that V has a value 27 0.6 m per second, keeping our units consistent. So that's the initial velocity of the stone. So now we might ask ourselves what is the initial height which the stone was stern to do that? We're going to want to figure out the change and why which is equal to the height change in life. So, let's see what we got where you can find the initial y velocity by doing visa. Y is equal to the times the sine of 25 degrees. Let's go ahead and do that using V is 27.6 m per second. Let's multiply it by the sign of 25 degrees to get that initial y velocity, we find that that is 11.6 six m per second. That is the Y component, the initial velocity of course, it's directed upward to find the final velocity which we can use later to figure out that distance. We're gonna use our first cinematic equation and say vis a vie final is equal to visa by initial plus the acceleration, which of course going to be minus g minus 9.8 m per second squared times the change in time. Of course we know that changed in time. It's 4.2 seconds. So this is going to be 11.66 m per second minus 9.8 m per second squared times 4.20 seconds 11.66 minus 9.8 times four point to find at that speed, that velocity is negative 29 point 50 m per second. Now, to finish this up, we'll say that Visa by final squared is equal to Visa by initial squared plus two times the acceleration which is minus G. Times change and why? So now to figure out the changing, why all we have to do is rearrange you'll say that a change in life if you put too. He said by final squared minus Visa by initial squared over two times minus plugging in our values, we're going to do the minus 29.50 squared minus 11.66 squared all over two times negative 9.8. We get a negative value and that value force that comes out to b minus 37 point. We'll go with five meters keeping our units consistent. So this value is the change and why? So when we ask ourselves what is the height equal to? H will say that that height is equal to 37.5 meters, the magnitude of that change and why? The last thing we want to figure out what is the maximum height reached by the stone? Well, for that we're gonna have the same Y initial speed 11.66 m per second. But now let's say that Visa by final is equal to zero. We're gonna use the exact same equation. Visa by final squared equals Visa by initial squared plus two times minus G tend to change. And why once again we're gonna solve for change of Y Except now we're going to say that Visa by final is equal to zero and keep her piece of my initial of 11.66 So into the calculator, I'm going to take Visa boy final, which is zero squared so we can completely forget that term. On top we have minus 11.66 squared and on bottom we have two times negative 9.8. And I'll go to you. Comes out to be a positive six point nine four meters. So the height from which or the maximum height reached by the stone is going to be the height from the ground. Plus this change in Y value. And so H max then is 37.5 m plus 6.94 m. We got a value of about 44 0.4 years. The maximum height reached by the stone. Thank you.

In this project, emotion problem would first have to find the height of a cliff, which the project is aimed at. To do that. We can use the following equation to track its horizontal or I'm sorry, it's vertical displacement. So it's uh that is going to be equal to the initial velocity in the y axis times of time -1/2 times its acceleration due to gravity times time. This is going to be the white component of velocity. So that's going to be 42 m per second times sine 60 degrees. We are told that the time is 5.5 seconds. So we know that we have all the information we need to solve for the high. Okay. Yeah. And this height gives us Almost 51.8 m. Now let's move on to part B where we want to find the speed and to find the speed at the point where it hits. We need both the wire and X components of velocity, so let's fine. The Y Component 1st and we can do that with the following equation. Mutual velocity in the Y axis minus acceleration due to gravity times. Time and again we use this component of velocity. Okay. Mhm. Thank you. And we use a simple gravity times the total time of flight, so about 5.5 seconds. And that's gonna give us roughly negative 17.5. And it makes sense for it to be negative because the projector is falling back down. So the vertical direction is going down right now the horizontal direction is going to be much easier to find because it doesn't change In project in motion. Darwin doesn't change. So we simply have to find the horizontal component of the initial velocity and that's going going to be 42 minutes per second times co signed of 60 degrees, Which is going to give us 21 m/s. Okay, No. To find the magnitude of the vector velocity We take the square root of negative 17.5 squared plus 21. Used for seven straight mhm. So that gives us the speed at that point is 27.3 meters per second. Now let's move on to part C in part C. We want to find this maximum height capital page and so we can use the same equation. But of course the time is going to be different. So let's write this down. Height maximum is going to be equal to the initial velocity in the y axis times time minus one half gravity times times for it. But we don't know the time. It took it together to get to that point. But we know that at that point it had to go from some velocity in the Y axis, 20 velocity in the y axis. So let's use that principle to find the time. We can say that the final velocity in the y axis, which we said is zero. It's equal to its initial velocity minus gravity times time. So we can solve for time and weekend at that time is equal to in the final velocity in the Y axis minus initial velocity in the Y axis. All over grabbing This one is 0 with them. We don't think about it. And then we have our initial velocity which we said was 42 times sine it's of 60 degrees all over negative 9.8 m per second because it is the X ray. This minus science has allowed And we get that time is roughly 3.7 seconds. So now we have all we need to solve bridge. And when we put this in a calculator We obtained the high is almost 67.5 m and that's our final answer.

Problem we have given a stone is thrown particularly downward from the roof of a building support. This is a roof of a building and height of this building is supposed edge and it is dropped, thrown vertically downward with some initial speed. Suppose that initial speed is V and it has given that it passes a window 16 m below 16 m below. Like this upon this distance is 16 m so 16 mitri below It has given that the speed is 25 m per second speed is 25 meter per second and it lands on the ground three seconds after it was thrown. What was initial velocity of the stone? And how tall is the building? So first of all, we have to find the initial velocity. We know that here a gravitational exploration is equal to G which is in the downward direction. So we will use equation. We finally is equal to the initial acceleration into time. No, here we will. We have not given the time. But we have given this displacement which is 16 m. So we will use. We finally squared is equal to we initially squared plus two a into s now we finally is 25. The square be initial. We don't know which is he go into isolation is minus 9.80 and displacement is minus 16 m because it isn't that downward direction. So a speed initial spirit will be equal to 25 square minus two into 16 and two, 9.80 Now we will solve this. So we will get the initial speed which will vehicle 225 mine 25 square minus 32 into 9.80 This is a quarter 17 point six meter for a second. This is the initial is fear of the stone in the vertical direction which is downward now if we talk about that, how tall is the building? It has given that it will take three seconds to land. So do you know that we will use here? Equation s is equal. Do you d plus half a T squared? Now? The vertical displacement will be equal to the height of building which is minus edge. An initial spirits 17.64 in the negative direction and time is 33 plus one by two. Exploration is minus 9.80 and time is three square. So this height of the building will be equal to 17.6. Multiply with three less 0.5 into 9.80 into nine, so this will be equal to 96 0.9 m. This is the height of building were just 96.9 weeks.

All right. So this question we have ah, boy throwing the stone and it's gonna fall down, right? And he's on a building. But we do not know the height of the building. And we're told that the angle of the initial velocity and we're told how far the stone lands horizontally. And we also told that this whole ordeal takes 4.20 seconds. So what do you have to do? Well, first things first. We hire asked to sketch graphs off four things. First, we have the exposition versus time. So that's just going to be linear line just like that, right? Because the velocity is constant. And so the rate of change of the position, the rate of change of position is gonna be constant. And so that's that. Next we have to sketch the graph of why the position of why in time. And so if, since it's falling down, words will start at the, um, start here, let's say and I am vote start graphic being something like this, right, Sapir Parabolic shape downwards. And that's just because the ah y acceleration is ah is present. So the Y velocity is constantly changing, but the y acceleration isn't, and so we're just gonna be a parabolic shape. Next, we have the the X graph with teeth, and that's pretty simple. Just gonna be a straight line across, right, Because the horizontal velocity is not changing, and then we have Okay, uh, then we have the vertical velocity versus time, and that's just going to be just something like that, all right, because the vertical velocity is changing, but it is changing at a constant rate because acceleration due to gravity is always going to be 9.81 meters per second. Right acceleration is always gonna be 9 21 for second. And so the rate of change of velocity is always gonna be the same since the linear graph. Cool, Cool. So now that we have these graphs, let's get to the question. Ah, first you got to figure out what is the initial velocity. Okay, so the easiest way to do this is to work at the horizontal component because it's gonna be seen throughout. We know that the thing, though, don't travels for 4.20 seconds and that it takes it travels 105 meters away. Right? So let's find the horizontal initial velocity. So that earthy horizontal loss in general, that's just going to be the distance over time. And that's gonna be 100 5 meters over 4.20 seconds, giving us a horizontal velocity of 25 meters per second. Okay, so since we know that this component is 25 minutes per second, then using trigonometry, we confined me Not because V not is just going to be equal to 25 meters per second. Divided by co signed 25 degrees, giving us 27.6 music second for vino. So there you go. Next, we have to find figure out the this height, right, the height of the building. How do we do that? Well, this is a much more complicated maneuver, but basically we should start thinking about the vertical component. Now that's all that matters. So let's find first off thieve vertical initial velocity. And so we could do that by going. We know it, um, sign 25 degrees and I was gonna give us 11.7 meters for sex. Cool. So that's sort of step one, then. Here's what I like to do is we're gonna find out how long it takes the ball to reach the peak. Okay? And we can do that pretty simply because we know at the peak obviously v wise gonna be called zero. So if we do an equation like this were vehicles V not plus acceleration T. Well, then this is equal to zero. So we have the not why is equal to negative of a, uh Y t and obviously accelerations. Negative anyways, but we have 11.7 are. Let's let yourself for time directly. We have time is equal to divert initial vertical velocity divided by the vertical acceleration. That's gonna give us 11.7 meters per second, divided by 9 21 meters per second squared. And so the time it takes for the peak right is going to be equal to one point about 1.19 seconds. So what do we do with this information? What what good does it do? Well, using the fact that this initial times 119 seconds, then we can figure out, uh, the height of this sort of this site. Right? Let me draw and let's say red using 1.19 seconds we can find out the height of the peak of the ball relative to its initial starting location. Okay, How do we do that? Well, it's pretty simple. We have the form of the Delta X is equal to V plus V. Not over two times t. Again, we know that he's gonna be equal to zero because the final velocity final vertical velocity is gonna be zero. So all we do is 11.7. Divide by 21.19 seconds. Give us an answer off. Um, 6.93 meters. Okay, This height, the height of the peak relative to its starting state is 6.93 meters. What do you do with that information? Well, we'll need it later. Next we got to do is realize that since this whole ordeal takes 4.20 seconds and two p, getting from the starting point to the peak takes 1.19 seconds. All the know if he's attracted to numbers, we can realize right that the time it takes from the ball to go from the peak all the way to the floor is going to take Ah, but 4.20 minus 1.19 It's going to be three point, dear. One seconds. OK, so this is the time to fall. Call it TTF. Okay, um and so this time to fall is useful because using it, we can figure out the total height from the peak all the way to the bottom. Okay, that's, uh, sort of, technically dancers to Dia's. Well, but what? We'll do this first. So how do we do that? Okay, well, we can use the kingdom attic formula of Delta X is equal to V, not tea plus 1/2 a. He squared. We know that the initial velocity at the people, the initial vertical velocity zero. So cross that out. So now we have dubbed the X is equal to 1/2 times 9.81 meters per second squared times, three points there, one seconds squared. And so if we put those numbers together, we get that the total height is 40 about 44.4 meters. Okay, so this is the height from the peak all the way to the bottom. And now, since we know that the distance between two peak and the initial starting 20.6 point three and we know that the total haIf from the four to the peak is 44.4 meters. Welcome. Subtract these two numbers to get our answer and so kind of running out of space here. But ah, if we read it here, will have 44.4 minus 6.93 gives us a ah, the height of the building as 37.5 meters. So I'll rewrite this year. Hi equals to 37.5 meters. And that is our answer to part C. And then party, we already calculated is going to be the height of the peak. And that's gonna be 44.4 meters. Right? So this is part C. This is part D, and there we have it.


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