Final answer:
The final speed of the 75-kg skier at the top of the slope can be determined by accounting for the work done by gravitational potential energy changes and friction. By using conservation of energy and the given initial speed, mass, height, slope angle, and coefficient of friction, we can calculate the skier's speed at the top of the slope.
Step-by-step explanation:
To determine the final speed of a 75-kg skier at the top of a slope, we need to consider the work done by nonconservative forces (i.e., friction) as well as the change in gravitational potential energy as the skier ascends a height of 3.58 m on a slope inclined at 9° above the horizontal. The skier's initial speed is 16.7 m/s, and the coefficient of kinetic friction between the skis and snow is given as 0.12.
The total work done on the skier is the sum of the work done by gravity and the work done by friction. We calculate the distance traveled up the incline by using the vertical distance (height) and the angle of the slope to find the hypotenuse of the right triangle formed by the skier's path.
Work done by gravity (Wg) is the change in gravitational potential energy, which is mgh, where m is the skier's mass, g is acceleration due to gravity, and h is the change in height. Work done by friction (Wf) is negative and calculated as the frictional force times the distance the skier travels along the slope. The frictional force is the product of the normal force (the component of the skier's weight perpendicular to the slope), the coefficient of kinetic friction, and the distance traveled.
To find the skier's final speed, we use the conservation of energy principle. The skier's initial kinetic energy plus the work done by nonconservative forces (gravity and friction) equals her final kinetic energy.