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A skier starts from rest at the top of a hill. The skier coasts down the hill and up a second hill, as the drawing illustrates. The crest of the second hill is circular, with a radius of $r=36 m$ . Neglect friction and air resistance. What must be the height $h$ of the first hill so that the skier just loses contact with the snow at the crest of the second hill?

18 $\mathrm{m}$

Physics 101 Mechanics

Chapter 6

Work and Energy

Work

Kinetic Energy

Cornell University

University of Michigan - Ann Arbor

University of Winnipeg

McMaster 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.

02:08

In physics, work is the transfer of energy by a force acting through a distance. The "work" of a force F on an object that it pushes is defined as the product of the force and the distance through which it moves the object. For example, if a force of 10 newtons (N) acts through a distance of 2 meters (m), then doing 10 joules (J) of work on that object requires exerting a force of 10 N for 2 m. Work is a scalar quantity, meaning that it can be described by a single number-for example, if a force of 3 newtons acts through a distance of 2 meters, then the work done is 6 joules. Work is due to a force acting on a point that is stationary-that is, a point where the force is applied does not move. By Newton's third law, the force of the reaction is equal and opposite to the force of the action, so the point where the force is applied does work on the person applying the force. In the example above, the force of the person pushing the block is 3 N. The force of the block on the person is also 3 N. The difference between the two forces is the work done on the block by the person, which can be calculated as the force of the block times the distance through which it moves, or 3 N × 2 m = 6 J.

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A skier starts from rest a…

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So let's start by making a free body diagram for this gear at this point. So we know that the downward force is mg. And then there's also a normal force and the some of these two forces should be equal to the centripetal force. So we can actually say that mg minus F N is equal to em. These quit over our but then we know that at the highest point, the skier just loses contact with the snow at the caressed off the second hill. What that means is that at the top of at the top of the crest of the second hill wth e f and will be zero, which gives me M G is equal to em the squared over R. And if I saw that equation for V that comes out to be equal to the root squared off our G and our is 36 g is 9.8, take the square root, so that comes out to be 18.0 meter per second. So what we do know is that at this point the speed is equal to 18.0 meter per second and now we can go back to the top off the first hill. And we're told that the initial velocity there is your because it says that the skier starts from rest at the top of the hill and now we can apply the Law of Conservation Off Energy, which says K not initial plus p not initial is equal to K Final plus p e Final. So the some off initial kinetic energy and final some off initial kinetic and potential energy equals the sum of final kinetic and potential energy. So we have 1/2 m v not squared, plus M g H is equal to 1/2 em times 18 squared plus zero can take the potential energy at the end point to be zero, cause that's lower. I want h from the first point so we can cross out all the EMS and we can now substitute our values are actually 1/2 m v, not squared will be zero. So we have G h is equal to 1/2 times 18 square substitute Jeez, equal to 9.8 salt for each. The answer comes out to be 18.0 meters per second

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