Two snowy peaks are at heights H=850 m and h=750 m above the valley between them. A ski run ex

step 1 So here we're just going to apply the law of conservation of energy. We know that initial potent

Two snowy peaks are at heights H=850 m and h=750 m above...

Two snowy peaks are at heights $H=850 \mathrm{~m}$ and $h=750 \mathrm{~m}$ above the valley between them. A ski run extends between the peaks, with a total length of $3.2 \mathrm{~km}$ and an average slope of $\theta=30^{\circ}$ (Fig. 8-61). (a) A skier starts from rest at the top of the higher peak. At what speed will he arrive at the top of the lower peak if he coasts without using ski poles? Ignore friction. (b) Approximately what coefficient of kinetic friction between snow and skis would make him stop just at the top of the lower peak?

Key Concepts

Normal Force on an Inclined Plane

On inclined surfaces, the normal force is the component of the gravitational force perpendicular to the surface. It is crucial for determining frictional forces since the kinetic friction is proportional to the normal force. The normal force on an incline is calculated as mg cos(?), where ? is the angle of the incline.

Kinetic Friction

Kinetic friction is the resistive force acting opposite to the direction of motion when two surfaces slide against each other. It is quantified by the coefficient of kinetic friction (µk) multiplied by the normal force. This frictional force does work that converts mechanical energy into thermal energy, effectively reducing the net energy available for motion.

Work-Energy Principle

The work-energy principle relates the work done by forces on an object to the change in its kinetic energy. When non-conservative forces (like friction) are present, their work must be taken into account, as they dissipate mechanical energy, reducing the kinetic energy that an object would otherwise have.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is calculated as the product of mass, gravitational acceleration, and height (mgh). Changes in height result in energy changes that drive motion, particularly when converting into kinetic energy during descent.

Conservation of Mechanical Energy

This concept states that in a closed system with only conservative forces acting, the total mechanical energy (the sum of kinetic and potential energy) remains constant. In problems involving gravitational motion without friction, energy lost from potential energy is completely converted into kinetic energy, allowing one to calculate speeds from height changes.

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