91Ó°ÊÓ

When a skier moves uphill, there is a force opposing this motion called the kinetic frictional force. This frictional force happens because the skis slide across the snow, and it acts in the opposite direction to the skier's movement. It's part of why the skier's speed decreases as they ascend the hill.
To figure out the work done by this force, we use the formula:

• \[ W_{\text{friction}} = f_k \cdot d \cdot \cos(\theta) \]

This equation means you multiply the frictional force \( f_k \) by the distance covered \( d \) and the cosine of the angle between the force direction and the displacement. Since the frictional force acts directly opposite the motion, the angle here is \(180^{\circ}\), making \( \cos(180^{\circ}) = -1 \).

• Kinetic friction is crucial in energy computations and affects the work-energy equation.

Understanding this concept helps explain why the skier's energy turns into heat due to friction.

As the skier moves uphill, gravity also plays a key role by influencing their potential energy. Gravitational potential energy (GPE) is the energy stored because of an object's position in a gravitational field. For the skier, climbing the slope involves gaining height, which increases their potential energy.
The potential energy change \( \Delta U \) reflects the energy transferred to an object as it moves upward:

• \[ \Delta U = mgh \]

Where \(m\) is the mass, \(g\) is the acceleration due to gravity (\(9.8 \, \text{m/s}^2\)), and \(h\) is the change in height. Here, \(h\) is determined by the vertical component of the skier's path, calculated using \( 1.9 \times \sin(25^{\circ}) \). This increase in GPE comes into play by reducing the skier's kinetic energy as they climb up.

Kinetic energy (KE) is the energy an object possesses due to its motion. As the skier moves up the slope, they start with an initial kinetic energy due to their initial speed (\(v_i = 6.6 \, \text{m/s} \)). As they move upward, the frictional force and the incline slow them down, leading to a decrease in kinetic energy.

• \[ \Delta KE = \frac{1}{2}m(v_f^2 - v_i^2) \]

Where \(v_f\) is the final speed, and \(m\) is the mass of the skier. This change in kinetic energy shows how the energy is transferred or transformed during the skier's motion up the slope. Precisely, it decreases by about \(-762.3 \, \text{J}\) as calculated.

• A decrease indicates that the skier is losing energy to both friction and potential energy gain.

This concept ties back to the work-energy principle, showing how energy conservation manages the skier's energy budget during the ascent.