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A ski that is placed on snow will stick to the snow. However, when the ski is moved along the snow, the rubbing warms and partially melts the snow, reducing the coefficient of kinetic friction and promoting sliding. Waxing the ski makes it water repellent and reduces friction with the resulting layer of water. A magazine reports that a new type of plastic ski is especially water repellent and that, on a gentle \(200 \mathrm{~m}\) slope in the Alps, a skier reduced his top-to-bottom time from 61 s with standard skis to 42 s with the new skis. Determine the magnitude of his average acceleration with (a) the standard skis and (b) the new skis. Assuming a \(3.0^{\circ}\) slope, compute the coefficient of kinetic friction for (c) the standard skis and (d) the new skis.

Step by step solution

01

Calculate Average Acceleration with Standard Skis

To find the average acceleration, we use the kinematic equation \( s = ut + \frac{1}{2}at^2 \) where \( s = 200 \text{ m} \) is the displacement, \( u = 0 \text{ m/s} \) (since the skier starts from rest), \( t = 61 \text{ s} \), and \( a \) is the acceleration. Rearranging gives \( a = \frac{2s}{t^2} \).Substituting the values, we have:\[a = \frac{2 \times 200}{61^2} = \frac{400}{3721} \approx 0.1076\] \( \text{m/s}^2 \)

02

Calculate Average Acceleration with New Skis

Repeat the formula \( a = \frac{2s}{t^2} \) for the new skis using \( t = 42 \text{ s} \):\[a = \frac{2 \times 200}{42^2} = \frac{400}{1764} \approx 0.2267\] \( \text{m/s}^2 \)

03

Compute the Coefficient of Kinetic Friction for Standard Skis

First, calculate the gravitational component along the slope using the formula \( g \sin(\theta) \) where \( g = 9.8 \text{ m/s}^2 \) and \( \theta = 3.0^{\circ} \). The net acceleration \( a_{net} \) is the result of the gravitational acceleration minus the friction:\( g \sin(3.0^{\circ}) - f_k \cdot g \cos(3.0^{\circ}) = 0.1076 \),Solving for \( f_k \) gives:\[f_k = \frac{g \sin(3.0^{\circ}) - 0.1076}{g \cos(3.0^{\circ})} \approx \frac{9.8 \times 0.0523 - 0.1076}{9.8 \times 0.998} \approx 0.046\]

04

Compute the Coefficient of Kinetic Friction for New Skis

Using the same method, the net acceleration \( a_{net} = 0.2267 \):\( g \sin(3.0^{\circ}) - f_k \cdot g \cos(3.0^{\circ}) = 0.2267 \),Solving for \( f_k \) gives:\[f_k = \frac{g \sin(3.0^{\circ}) - 0.2267}{g \cos(3.0^{\circ})} \approx \frac{9.8 \times 0.0523 - 0.2267}{9.8 \times 0.998} \approx 0.030\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

average acceleration

Average acceleration is a fundamental concept in physics, describing how quickly an object's velocity changes over time. When skiing down a slope, the acceleration can vary based on several factors, including friction and the angle of the slope. To determine the average acceleration accurately, we use certain parameters.

• Initial Velocity \(u\): Since the skier starts from rest, the initial velocity \(u\) is \(0\) m/s.
• Displacement \(s\): This is the distance traveled by the skier, which in this case is \(200\) meters.
• Time \(t\): The time taken to travel \(s\), different for standard and new skis, i.e., 61 seconds and 42 seconds respectively.
• Acceleration \(a\): Using the kinematic equation \(s = ut + \frac{1}{2}at^2\), you can find \(a\) by rearranging it to \(a = \frac{2s}{t^2}\).

Substituting the values for standard skis, the calculation shows \(a \approx 0.1076\) m/s², and for new skis, \(a \approx 0.2267\) m/s². This change indicates the reduced friction and improved acceleration due to the enhanced ski materials.

kinematic equations

Kinematic equations are essential for calculating the motion of objects when acceleration is constant, such as a skier moving down a slope. These equations relate an object's displacement, initial velocity, time, and acceleration, allowing us to find unknown variables when certain conditions are known.

One of the key equations is \(s = ut + \frac{1}{2}at^2\). This equation can be particularly useful in motion analysis where the initial velocity \(u\) is zero, meaning the object starts from rest:

• \(s\) (Displacement): Represents the distance covered by the object.
• \(u\) (Initial Velocity): For a skier starting from rest, this is \(0\) m/s.
• \(t\) (Time): The duration over which the motion occurs.
• \(a\) (Acceleration): This can be calculated by rearranging the equation, \(a = \frac{2s}{t^2}\).

These equations provide a clear framework for solving problems related to uniform acceleration, a common scenario during skiing.

gravitational component

The gravitational component plays a vital role in the motion of objects on inclined surfaces, such as a skier sliding down a slope. When an object is on a slope, gravity acts in two significant directions: parallel to the slope and perpendicular to the slope.

• Parallel Component \(g \sin(\theta)\): This portion of gravitational force impacts the motion down the slope and influences acceleration.
• Perpendicular Component \(g \cos(\theta)\): This modifies the normal force and hence affects the friction experienced by the skier.

In the ski example, the slope angle, \(\theta = 3.0^\circ\), allows for the calculation of the gravitational component through \(g \sin(3.0^\circ)\). This contributes significantly to the net force acting on the skier, and when combined with frictional forces, determines the skier's acceleration and speed.

coefficient of kinetic friction

The coefficient of kinetic friction \(f_k\) quantifies the resistance between two surfaces sliding past each other. For a skier, this is the friction between the skis and the snow.

• Low \(f_k\) indicates less friction and smoother motion.
• High \(f_k\) indicates more resistance and slower speeds.

To find \(f_k\), we assess the forces in play. The net acceleration \(a_{net}\) is calculated by considering the gravitational component and the friction:
\(g \sin(\theta) - f_k \cdot g \cos(\theta) = a_{net}\)
By rearranging this formula to solve for \(f_k\):

• For standard skis, \(f_k \approx 0.046\).
• For the new skis, \(f_k \approx 0.030\).

These calculations illustrate how the new plastic skis have a lower coefficient of kinetic friction, facilitating smoother and faster descent compared to standard skis.