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(11) (a) A long jumper leaves the ground at $45^{\circ}$ above the horizontal and lands 8.0 $\mathrm{m}$ away. What is her "takeoff" spced $v_{0} ?(b)$ Now she is out on a hike and comes to the left bank of a river. There is no bridge and the right bank is 10.0 $\mathrm{m}$ away horizontally and $2.5 \mathrm{m},$ vertically below. If she long jumps from the edge of the left bank at $45^{\circ}$ with the speed calculated in $(a),$ how long, or short, of the opposite bank will she land (Fig. 43$) ?$

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a) 8.9 $\mathrm{m} / \mathrm{s}$b) 10.1 $\mathrm{m}$

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

Cornell University

Hope College

University of Winnipeg

Lectures

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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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our problem, says a. A long jumper leaves the ground at 45 degrees above the horizontal and land eight point zero meters away. What is her takeoff Speed V? Not that's part a. So let's do that first before we go on in salt Part B. So what have we told for part? A. Well, we're told that her angle is they do not okay, equal the 45 degrees. All right. We're also told that the range in which she travels will call that R is equal to eight meters. Okay. And the question is, what is her velocity? Her initial velocity. All right. In order to do this, we can go ahead and make use of the range formula which says that the range is equal to the initial velocity time's sign of two times the initial angle, which is state or not divided by gravity. Okay, so if we solve here, I'm sorry to be not here is actually squared, so it be not square. Okay, So if we solve here for V not we find that Vietnam is equal to the square route because you have two square root each side to get rid of this squared our times G divided by sign to fate or not. Okay, no. We have all of our values here that we would need to use for this. The range we were told actions ahead. The square root is eight meters right and you're multiplying that by gravity which is 9.8 meters per second. Swear because its acceleration divided by signed to Fada Fada is 45 degrees, right? So sign Earth. So two times 45 degrees is 90 degrees and sign of 90 degrees is just for one. So this is going to be divided by one. You carry out this calculation, it's right down here. You will find that this approximately equals 8.9. Hey! And that's meters per second because its velocity. And that's our solution for part a box at it. Okay, Now for part B, it reads. Now she is out on the hike and comes to the left bank of a river. There is no bridge in the right bank is 10 meters away horizontally and 2.5 meters down vertically. If she long jumps from the edge of the left bank of 45 degrees with the speed calculated today. How long or short of the opposite vanquished he land like and figure 43. OK, so let's go ahead and open up a new tab here and well, do this part as part B label is part B. Okay? And for part B, we are told that the distance she's trying to travel is 10 meters away. So we'll say X is equal to 10 meters. Okay. And the bank that he was trying to tap travel is down 2.5 meters from where she's jumping. So we'll say minus 2.5 years because it's down. And so that means her initial point is why zero equals X zero equals zero. So she's at the origin to start with, okay? And we're also told that her velocity Vina is ah the same as thie velocity that we calculated in the first part which was 8.9 meters per second. Okay, so now this tells us that, uh, what we can use the equation. Why is equal to why not? Plus Vina Times the time plus 1/2 a t squared acceleration in the white direction Times t squared, OK, And the reason we're going to use this because we want to solve for the time. So we need to know how long she's in the air in order to know if she's going to make it that full 10 meters, that she needs to make it in order to know how long she's in the air. We're going to need to figure out what hope her launching. I'm sorry. We're going to figure out what? How much time? She's in the year vertically. And if you know how much time she's in the air vertically, we're gonna know how much time she is in the air horizontally. Okay, well, her initial velocity is vey so. We need Veena. Why? And I don't think I included No, I did not. I did not include the Why here We need V not why, Okay. Do not. Why is equal to Vina times the sine of the angle data which is 45 degrees. We have the same angle, this one which were also told. Okay, So if we plug in all of our values here we find that this is equal to why which is minus 2.5 meters. Oh, which is equal to why not? Which was zero. So we can ignore that. And this is 8.9 times. This is a meters per second keeper units here times the sign of 45 immunised agrees. What time's the time? Plus and this is 1/2 multiplied by gravity because acceleration here for us is gravity. So it's minus 9.8 meters per second squared, right, and then this is multiplied by t squared, okay. And t squared is what we're trying to find. We have multiple values here that don't have tea, But let's go ahead and put all the values on one side of the equation and make this equal to zero. And if we do that, we find that this is 4.9 t squared. Okay, 4.9 is 9.8, divided by two. And this is going to be minus 6.261 t right? And that comes from 8.9 times signed 45 degrees and then this is going to be minus 2.5. All this is going to be equal to zero. Okay, Now we can use the quadratic formula to solve for tea, and if we do that. We find that tea then is equal to, uh, 6.261 because it's minus minus 6.261 plus or minus square root of 6.261 minus four times 4.9 multiplied. Bye. Minus 2.5. And then all this is going to be divided by two times 4.9. What's the quadratic formula there that we're using? You carry this out, you'll find that you're gonna get two values, Of course, because of that, plus or minus. And this is equal to one of them is minus 0.319 And the units on Mrs Seconds. Of course, because it's time and the other one is one 0.5975 course again. Unit seconds. Because it's time. Okay, well, of course you can't have negative time. We're not going to use the negative times. We're going to use the one That's the positive time. And we're going to get to use that to calculate the distance, traveled in the ex direction. Shit. Okay. And we can do that here. We're in a new page once again. Uh, we could do that by saying the distance travelling. The extra action. We're gonna call this. We'll just call this ex prime for the distance traveled is equal to the velocity in the extraction. Right? Time's the time traveled. Okay, well, the velocity and extraction is equal to the initial velocity Veena Times that co signed of the angle which is 45 degrees okay. And V knock was again 8.9. Oh, and this is multiplying. Still buy tea, of course. 8.9 meters per second. Times co signed 45 multiplied by t which if we go back to the previous page, we see that tea is 1.5975 So this is multiplied by 1.5 nine seven five and the units on that one is seconds. Okay, Carry the south of seconds. Cancel and you left with meters and you find that this is equal to 10 meters. Therefore, X crime is equal to the X, which is the distance the bank was away. Thus well, type this out. The solution then it is. She lands perfectly on the bank and this box that in hey as our solution. Okay,

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