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In the annual battle of the dorms, students gather on the roofs of Jackson and Walton dorms to launch water balloons at each other with slingshots. The horizontal distance between the buildings is 35.0 m, and the heights of the Jackson and Walton buildings are, respectively, 15.0 mand 22.0 m. Ignore air resistance. (a) The first balloon launched by the Jackson team hits Walton dorm 2.0 s after launch, striking it halfway be- tween the ground and the roof. Find the direction of the balloon’s initial velocity. Give your answer as an angle measured above the horizontal.(b) A second balloon launched at the same angle hits the edge of Walton’s roof. Find the initial speed of this second balloon.

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(a) angle is $24.02^{\circ}$ (b) launch speed is 28.92 $\mathrm{m} / \mathrm{s}$ which makes sense because it's greater than in earlier case

Physics 101 Mechanics

Chapter 3

Kinematics in Two Dimensions

Motion in 2d or 3d

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Lectures

04:01

2D kinematics is the study of the movement of an object in two dimensions, usually in a Cartesian coordinate system. The study of the movement of an object in only one dimension is called 1D kinematics. The study of the movement of an object in three dimensions is called 3D kinematics.

10:12

A vector is a mathematical entity that has a magnitude (or length) and direction. The vector is represented by a line segment with a definite beginning, direction, and magnitude. Vectors are added by adding their respective components, and multiplied by a scalar (or a number) to scale the vector.

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So there are two parts of this question. The first part of the question. Ask what angle the water balloon has to be thrown at if it's thrown from the 15 meter building to the 22 meter building and it lands at halfway between the buildings at 11 meters. Um, and it takes the water balloon two seconds to get there, so find the angle it's thrown at here. We just need to use our chemical equations to solve for the components of the velocity vector here. So let's first focus on the vertical component of the velocity so we can use the kinetic equation, which states that the change in the why displacement is equal to the initial velocity in the Y direction times the time plus 1/2 times the acceleration times, the Times Square. So we know that displacement in the Y direction is going to be negative four meters because it starts at 15 meters and ends up at 11 meters negative, for we don't know what the vertical velocity is. We do know what the time is. It's two seconds to get there, plus 1/2 times the acceleration, which is negative. 9.8 times the time again squared, which is two seconds to square. So when we simplify this and so for visa, why by it's attracting 1/2 times negative 1.8 times two square on both sides and dividing by two we get the vertical component of the velocity is equal to 7.8 meters per second. And so now that we have this, we can ah find what the horizontal component is of the velocity vector. And to do this it's pretty simple. We know that the equation that discouraged the range of the Project O is the same thing as the initial velocity in the horizontal direction times the time is equal to the range, and so we know what the time is, and we also know what the range is. So we console for the visa, Becks, So six of X a times two is equal to 35 meters the distance between the buildings, and we find that be sub X is equal to 17.5 meters per second. So now we have our the velocity vector here that we're trying to find. We know the horizontal component is 17.5 meters per second and we know that the vertical component is 7.8 meters per second. And so to find the angle here, all we have to do is take the 10 inverse, which is opposite over adjacent. So 10 inverse of 7.8 divided by 17.5. And that will give us the angle Arteta, which will be equal to 24 0.2 degrees. So that's the answer to the first part of the problem now to answer the second part of the problem where asked to find what the speed is of the water balloon if it shot from the same angle. But this time it lands right on the edge of the other building. So to do this, we're going to need to find the time in terms of the philosophy. So let me make a new page. So we know we're trying to find this velocity here. V. We know the angle here is 24.2 degrees, and this means that our horizontal component is going to be equal to B Times Co sign of 24 0.2 degrees because co sign is adjacent over I upon news and the adjacent is Visa X. So Visa Vex Weekly v. Times Co sign of 24.0. So, um, if we want to find the time in terms of this velocity V, we can say that the range of the water balloon which we he's talked up for is just d, uh, horizontal component of los T times. The time is equal to the range, so we can plug in what we knew. So our horizontal component is V. Times Co. Sign of 24.2 degrees. Times T is equal to 35 meters, so we know that T is equal to 35 divided by the co sign of 24 0.2 degrees. Now that we have our tea, in terms of this velocity V, we can look at the vertical component of the velocity, which is just gonna be a V time sign of 24 0.2 degrees. And we can use our Kinnah Matic equation which states that the change in displacement in the vertical direction eyes equal to the initial velocity in the vertical direction times the time plus 1/2 times the acceleration times, time squared. So we know our displacement in the vertical direction is going to be 20 to minus 15 meters. So seven meters, seven meters is equal to the initial lost E the time sign of 24 0.2 degrees times the time. So our time that we got up here Times 35 divided by Sorry, let me rewrite. That's come. Messi 35 divided by the Times Co sign of 24.2 degrees plus 1/2 times the acceleration, which is negative. 9.8 due to gravity. Times are time squared so 35 over the co sign, 24 0.2 degrees squared. And so, um, we can simplify this of these V's cancel out here and also the sign over co sign turns into tangent. So we get seven is equal to 35 times he tangent of 24.2 degrees plus 1/2 of 9.8 is 4.9. So negative 4.9 times 35 over the co sign, uh, 24 0.2 degrees squared. And now, since we only have one V term left, we can solve for this term so we can subtract 35 times 10 of 24.2 degrees on both sides. Then we can defied by negative 4.9 on both sides. Take the square root. And then, um we can multiply by V on both sides and divide by whatever term we had still on the left side to get toe isolate V by itself. Um, and when we do this, we find that the philosophy is equal to 28 0.92 meters per second and that's the final.

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