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(II) At what projection angle will the range of a projectile cqual its maximum height?

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$76^{\circ}$

Physics 101 Mechanics

Chapter 3

Kinematics in Two or Three Dimensions; Vectors

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Rutgers, The State University of New Jersey

Simon Fraser University

University of Sheffield

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In physics, a rigid body is an object that is not deformed by the stress of external forces. The term "rigid body" is used in the context of classical mechanics, where it refers to a body that has no degrees of freedom and is completely described by its position and the forces applied to it. A rigid body is a special case of a solid body, and is one type of spatial body. The term "rigid body" is also used in the context of continuum mechanics, where it refers to a solid body that is deformed by external forces, but does not change in volume. In continuum mechanics, a rigid body is a continuous body that has no internal degrees of freedom. The term "rigid body" is also used in the context of quantum mechanics, where it refers to a body that cannot be squeezed into a smaller volume without changing its shape.

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In physics, rotational dynamics is the study of the kinematics and kinetics of rotational motion, the motion of rigid bodies, and the about axes of the body. It can be divided into the study of torque and the study of angular velocity.

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(II) At what projection an…

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our question says that at what projection angle will the range of a projectile equal its maximum hike? So, OK, in order to do this, we're going to start by choosing the origin to be where the projectile is launched and upward to be the positive Y direction. The initial velocity of the projectile is we're going to call V of zero, the launch angle, a state A zero. Right. So we're trying to find data zero and someone go ahead and write that down. We want to know what they zero equals. Okay, Well, um, in order to do that, the range of the projectile is given by the range formula. So let's go ahead and start out with the range formula because the range formula has stayed a zero in it. Well, that says that the range are is equal to ve not squared time. Sign of tooth. They did not. They did not being what we're trying to find, okay. Designed by g gravity. Okay, but we don't know way. Don't know. V not squared, right? And we also don't know the range. But what we do know is that the range, um or that why is maximum. Yeah, um, or wind my hazmat maximum. The veena value. Why? Max means that dizzy not. Or the V of live value here is equal to zero. Because at the very top of that motion, if you launch something straight up in the air right before it starts to come back down, it has a velocity of zero. Started to maximum height if lost, year zero. Okay, so then we can go ahead and use the equation. V. Why Squared is equal to the wives. Zero squared, plus two time's a of y times. Why, Max, Tell me. Call Weiss of him. Minus why of zero? Well, v of lie. Because we are dealing with why Max zero zero, right? And why of zero is zero. Okay, so we can ignore those. And we're going to end up with Zero is equal to the the Y of zero ve y of zero is equal to zero times signed data. So the wives zero squared is vo zero squared times signed data squared, and then we're gonna subtract from that too. I'm sorry. We're going to add to that two times a of why, which is minus g, because that's the gravity and the negative light direction. Right? Time's why, Max? Why? Sit in. Okay. So now let's go ahead and solve for y Max. That's equal to be not squared. So again, we have v Nandan are equation We not squared and signed data. I'm sorry we have v not squared in our equation. We also have stayed and not in our equation, Which is again, what we're trying to solve for and we have to g Okay. But we notice that, um, we want to find the angle for which the range is equal to Why maximum. So we're going to set the range equal. Why maximum? If we do that, we can set these equations equal to one another. Okay. And why the range was V not squared. Sign of two things or not divided by G. This is going to be equal to why Max, which was V not squared. Mr. Chris's should be squared over here. Times sine squared. They do not divided by to G. Okay, well, we'll notice that there are a couple of things that cancel here. I'll put those in red. The V not squares. Cancel and the gravity cancels out Okay, so now let's write down here. What? This leaves us. This is sign of tooth. Ada, not Jeffrey. Out to write. The not up here is equal to Well, we can multiply the two over under this side so you can say two times. Sign Tooth. They did not. Is equal to science where they do not. Okay. Well, using some trigger demetrick identities, we can write sign tooth Ada as two times Signe. I'm sorry. Signed Tuesday to not as two times signed data not co sign data. Not so that two gets multiplied by the two out front. The tardy bare sand up with four time's sign. They did not co sign tonight. Okay? And that's equal to science. Where they did not. Okay, well, now the sign they did not on one side is going to cancel out the squared on the other side. Okay. And let's give new paychecks. We ran out of room on this page. What we're going to have is four co signed it, and I is equal to scientist enough, okay? And again, we're trying to solve for they did not well signed data over co signed data is equal to tangent data. So we see that tangent data is equal to four. Since scientist over coastline data equals change in data. Okay, now it's all for things or not. They did not equals attention to the minus one, which is inverse tangent of four. The sequel 76 degrees your inbox set in as their solution.

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