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A hunter aims directly at a target (on the same level) 68.0 $\mathrm{m}$away. (a) If the bullet leaves the gun at a speed of 175 $\mathrm{m} / \mathrm{s}$ ,by how much will it miss the target? $(b)$ At what angleshould the gun be aimed so the target will be hit?

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a) $0.740 \mathrm{m}$b) $0.623^{\circ}$

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

University of Washington

Simon Fraser University

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Okay. Our question says that a hunter aimed directly at a target on the same level that is 68 meters away. Then a If the bullet leaves the gun at a speed 175 meters per second, by how much will it miss the target and be at what angle should the gun be aimed so that the target is hit? Okay, So I wrote down what we were given here, both directly and indirectly. And that is that the distance which I write his ex that the target is a 60 meters, the velocity and the ex direction. Since the hunter fires in the direction of the target is 175 meters per second, there is no velocity in the white direction because he shoots directly at the target on. We're going to assume that the target, which is at the same level as the barrel of the gun, is our starting point for why. And that's going to Weiss of zero minutes. Call that zero right. We're going to say that's at the origin and we wanted so clearly it's going to drop them because of gravity. We want to know how much it's going to drop over that distance, and so far how far away from the target it's going to be because of that drop. And so, in order to do that, we can go ahead and make use of the formula. Why is equal to why not plus the initial velocity in the white direction via y zero times the time, Okay. And, uh, since why is in units of distance via why time's the time also give you units of distance. So we also will have plus 1/2 acceleration in the wider action Times t squared again. Acceleration is meters per second squared times t squared, get you back to units of distance. So we have the correct units in all of these cases. Okay, so, uh, we have in this equation why of zero. We already know, though, that we have v Y zero. We know a of lies. The acceleration in the UAE direction is gravity, right? So and why is what we're trying to find? The only thing we don't have yet is time, but we have all the values we need to find the amount of time it's going to be in here because X is 68 meters, right and ah v ahs 125 meters per second. Well, the distance something travels is equal to the velocity that is travelling at times the time. So the time it takes for it to get to the target in the time it will have in order to drop a certain distance is going to be that time t. So if we saw for T year, we have tea is equal to X, the distance divided by V of X, the velocity and next direction Okay, which is going to be equal to 68 meters divided by 125 meters per second. So that's going to give us the units of seconds. Hey, if you do this calculation, you find this comes out to be zero 0.3 886 units of seconds. Okay, so now we can plug all the things we know into this equation for why we consult for why, by the way, this is part ay that we're working on here. So let's to note this is party. So why is equal to why not a zero? The of y 00 times is zero plus 1/2. Okay. A of why? Well, uh, that's minus 9.8 meters per second squared, and then we just found t right. That was zero 0.386 seconds, and that's also squared. Okay, so if you do this, you find this comes out to B minus zero. Quaint 740 meters. So it lands 0.740 meters below the target. That's how far, Mrs Abi. Okay, so now the question is, what angle would we need to use in order for it Not to miss the target by that much. Okay, well, we know the range it travels, right? That's Ah X. So we could use the range formula here to find the angle. Because in the range formula, we have angle. And we know everything else in this formula except for angle. So the formula for range is our is equal to be not squared. Sign of tooth data divided by gravity. Okay, so we want to solve for theta. So let's leave the sign to Feydeau. Let's get try to isolate Data is equal to R crimes. G divided by you have zero squared. OK, so now to get data by itself, We would need, um excuse me to get there by itself, We need to take the inverse side of both sides and then divide by two. So we take the universe sign of those sides, Something for data. So the sign on the left side cancels out and you have sign minus one. This is our times G divided by be, not squared, right. And then you have the two fatal. So this all has to be divided by two secrets to the 1/2 out front here. Okay, so now if we plug in all of the values that we have, we still have 1/2. You have inverse signed. Our is the range that's going to be 68 g is gravity acceleration 9 20 and that's in meters per second squared. But 68 is in meters. And then, of course, this is going to be a divided by Vienna squared, which is meters per second squared, most squared. So all the units we're gonna cancel out so we can just ignore units. So this is going to be 175 squared. Okay, You put all that in your calculator and this comes out to be zero 0.6 23 degrees, and that would be the angle that the hunter would need to shoot in order to hit the target from that distance away.

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