In downhill speed skiing a skier is retarded by both the air...

In downhill speed skiing a skier is retarded by both the air drag force on the body and the kinetic frictional force on the skis. (a) Suppose the slope angle is $\theta=40.0^{\circ}$ the snow is dry snow with a coefficient of kinetic friction $\mu_{k}=0.0400,$ the mass of the skier and equipment is $m=85.0 \mathrm{kg}$ , the cross-sectional area of the (tucked) skier is $A=1.30 \mathrm{m}^{2}$ , the drag coefficient is $C=0.150$ , and the air density is 1.20 $\mathrm{kg} / \mathrm{m}^{3} .$ (a) What is the terminal speed? (b) If a skier can vary $C$ by a slight amount $d C$ by adjusting, say, the hand positions, what is the corresponding variation in the terminal speed?

Key Concepts

Air Drag Force

Air drag is a resistive force that depends on the object's velocity, drag coefficient, cross-sectional area, and the density of the air. This force increases with the square of the velocity, providing a significant retarding effect at high speeds, and is crucial in determining the terminal velocity of an object moving through the air.

Variation of Terminal Speed with Drag Coefficient

The terminal speed is sensitive to changes in the drag coefficient, which characterizes the effectiveness of air resistance. A slight variation in the drag coefficient can lead to a corresponding change in terminal speed due to the drag force's quadratic dependence on velocity. This concept is important in performance optimization, where small adjustments in body position or equipment can impact the overall speed.

Force Equilibrium on an Incline

In the context of inclined motion, force equilibrium refers to the balance between the component of gravitational force parallel to the incline and the resistive forces, namely kinetic friction and air drag. When these forces are balanced, the object reaches terminal speed. Understanding this balance is essential for analyzing motion on slopes.

Terminal Speed

Terminal speed is the constant speed an object reaches when the force of gravity is exactly balanced by resistive forces such as air drag and friction. At this speed, the net acceleration becomes zero, and the object continues to move at a steady velocity down an incline or through a fluid medium.

Kinetic Friction

Kinetic friction is the force that opposes the motion of an object sliding over a surface. It is typically quantified by the product of the normal force and the coefficient of kinetic friction. On an inclined plane, friction reduces the net force available for accelerating the object down the slope and must be accounted for when calculating terminal speed.

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