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$\bullet$ $\bullet$ A sled with rider having a combined mass of 125 $\mathrm{kg}$travels over the perfectly smooth icy hill shown in theaccompanying figure. How far does the sled land from thefoot of the cliff?

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$x=25.54 \mathrm{m}$

Physics 101 Mechanics

Chapter 7

Work and Energ

Physics Basics

Applying Newton's Laws

Kinetic Energy

Potential Energy

Energy Conservation

Cornell University

Rutgers, The State University of New Jersey

Simon Fraser University

Lectures

03:47

In physics, the kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest. The kinetic energy of a rotating object is the sum of the kinetic energies of the object's parts.

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

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So this is a very interesting problem here. We'll see that the use ofthe conservation, off energy and also the question of projectile motion to solve this problem. So what we need to do is finally we need to find out the amount the distance of slate travels after it leaves the cliff. Now to do so. First, we need to find out the initial velocity or the velocity of the sled at this point. So if we consider this part of the projectile, that's going to be the initial velocity of the slope. If we consider just this part of projectile. But if we consider this part the left side of the figure, then this will be the final velocity of the sled. If 22.5 meter per second is the initial velocity now the salt, or to get the velocity at this point, um, so let's actually quality final because we're just dealing with this part now and call this part of the initial. So to find out the final the last you will be using conservation off energy and ah, once we get the velocity will do the projector. So let's do let first final DF then now from conservation of energy. We know that Ah, the potential energy plus the kinetic energy at final point should people to the potential energy and the kind of the energy at the initial point. So let's call this point as the initial point at this point as the final point. Now in this point Ah, If we consider this as our origin in the white direction, then why will be zero? And why f at this point why f will be 11 meters? So at the initial point, there's no potential energy. All we have is kind of big energy, so half and the I squared. So let's meet this figure a little bit smaller so that we have more room to work with. Now that is equal to defining kind of energy at the final point, which is M times we have squared plus the potential energy, which is m times she times. Why f so we can get you off the arms here. And our goal is to find VF from there. So if we solve for VF, we get B I squared minus two g h. And then if we can use the numbers we see that the eyes 11. 22.5 meter per second then Rivera square time minus two times 9.8 meters per second Spread times 11 meter. So this age is Ah fear. Why f here? So let me actually right Why f here toe just to avoid confusion. So that is why I f now using that we get the f as 17.1 meters per second. So now let's concentrate on the right side off the image. Now here we have the ah, When this lady is here, then it has 17 point 1,000,000 per second as its fun initial velocity and at this point will have the final velocity. So will be using We can So let's actually first draw the projector. Then we can discuss what to do with it. Right? So now this Let's call this part as the initial velocity and ah, let's call this as y f again so we can divide this projector. The question into two parts one is in the white direction. One is in the extraction. So from the wind direction or the part of vertical component of the motion will find the time and ah, we'll find the time that the sled that this lady is in the air and then use that time in the horizontal component to find the horizontal displacement. So the questions will be why f which is equal to I mean, I Why t plus half off e Y p squared. And for x f it's the eye x times t plus half off ext Time t spirit. So Ah, now this time, this won't be Why, If this will rewire Ah, this is the initial point. So this part, it's not why this is the way of position. So what if we see that? So now, in this case, let's actually considered this as our origin. That means ah, why initially, zero. At that point, that means why f will be negative 11 meters. So using that we can solve for tea which is so we'll be using this equation. First 13 it is squared off two times y f divided by anyway. Now you see that we're ignoring this part because at the initial point, the velocities perfectly tangential tow the project in which is prepared, which is parallel to the horizontal direction. That means there's no component in the vertical directions, and we ignore this part. So using this, we can solve for the time, which is two times minus 11 neater, divided by the acceleration, which is the grabbed, its gravitational acceleration. And since we're considering the upward direction as positive direction, then anything that that goes under ah discordant or which is, which is the direction opposite to the positive direction is negative, for it is negative. 9.8. I need every second squid. So from here we get time as 1.5 seconds. So using that time over here we can solve for Except so now this quantity right here is this guy, because when the ladies around here, the all the direction of the velocities in the horizontal directional Palito horse in a direction, that's why we'll have the velocity. We'll have this velocity as the I X tee times t. So from there, if we saw for Ex F Time's excess, we'll have the i X times D plus half off X t squared. Now this time will be ignoring. This part is zero. And the reason for that is because in this direction there's no exploration. All The acceleration that we have here is due to the gravitational acceleration which is acting downwards. So there's no X elation in this direction, so we can just cross it. That means this part goes to zero. So using that relation over here, we know that V I is B I x is 17.1 meter per second and tying we calculated as 1.5 seconds Using that, we find X f as 25.6 meters. So this distance right here is 25.6 meters. Thank you.

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