Question

A snowboarder starts from rest, rides \(38.09 \mathrm{~m}\) down a snow-covered slope that makes an angle of \(30.15^{\circ}\) with the horizontal, and reaches the flat snow near the lift. How far will she travel along the flat snow if the coefficient of kinetic friction between her board and the snow is \(0.02501 ?\)

Short answer

In this problem, we have a snowboarder sliding down a slope and then along the flat snow. The slope has an angle of 30.15 degrees and the snowboarder travels 38.09 meters along the slope. The coefficient of kinetic friction between the board and the snow is 0.02501. We are asked to find the distance traveled along the flat snow before the snowboarder comes to a stop. Using trigonometry, conservation of mechanical energy, and the work-energy theorem, we calculated the initial height of the snowboarder as 19.1 meters and her velocity on the flat snow as approximately 19.52 m/s. The work done by friction was calculated to be equal to the initial potential energy, which allowed us to find the distance traveled along the flat snow. The final answer is that the snowboarder will travel approximately 763.29 meters along the flat snow before coming to a stop.

Step-by-step solution

Identify the relevant physical quantities

We will primarily be dealing with the following physical quantities: - Initial height of the snowboarder (h) - Distance traveled along the slope (d_slope) - Angle of the slope (theta) - Coefficient of kinetic friction between the board and the snow (mu_k) - Mass (m) and acceleration due to gravity (g) are also involved in the calculations but their values will cancel out eventually, so we don't need their exact values.

Calculate the initial height of the snowboarder

We can use the angle of the slope and the distance traveled along the slope to find the initial height `h` using trigonometry (sin(theta) = opposite/hypotenuse): $$ h = d_{slope} \cdot \sin{\theta} $$ Plugging in the values, $$ h = 38.09\mathrm{~m} \cdot \sin{30.15^{\circ}} \approx 19.1\mathrm{~m} $$

Calculate the initial potential energy and kinetic energy on the flat snow

At the top of the slope, the snowboarder has potential energy due to her height (h): $$ PE_{initial} = mgh $$ When the snowboarder reaches the flat snow, she has no potential energy left, but she has gained kinetic energy (KE) given by: $$ KE = \frac{1}{2} mv^2 $$ Since the mechanical energy is conserved, we can equate the initial potential energy and the kinetic energy on the flat snow: $$ PE_{initial} = KE $$ $$ mgh = \frac{1}{2} mv^2 $$

Calculate the velocity on the flat snow

Now we will find the velocity `v` on the flat snow. From conservation of energy expression above, we can find `v`: $$ v = \sqrt{2gh} $$ Plugging in the height (h) we found earlier: $$ v = \sqrt{2 \times 19.1\mathrm{~m}} $$ $$ v = \approx 19.52\mathrm{~m/s} $$

Calculate the work done by friction

The work done by friction force is given by: $$ W = f_{friction} \cdot d_{flat} $$ Where `f_friction` is the friction force acting against the snowboarder's motion, and `d_flat` is the distance traveled along the flat snow, which we want to find. The friction force can be calculated as `f_friction = mu_k * m * g`, so: $$ W = (\mu_{k}mg)d_{flat} $$ Using the work-energy theorem, we can relate the work done by friction to the change in kinetic energy: $$ W = KE_{initial} - KE_{final} $$ Since the snowboarder comes to rest, \(KE_{final} = 0\), and we know \(KE_{initial} = mgh\). So: $$ W = mgh $$

Calculate the distance traveled along the flat snow

Now we can equate the expressions for the work done by friction and solve for `d_flat`: $$ (\mu_{k}mg)d_{flat} = mgh $$ The mass (m) and acceleration due to gravity (g) cancel out: $$ d_{flat} = \frac{h}{\mu_{k}} $$ Plugging in the values for `h` and `mu_k`, we obtain: $$ d_{flat} = \frac{19.1\mathrm{~m}}{0.02501} \approx 763.29\mathrm{~m} $$ The snowboarder will travel approximately \(763.29 \mathrm{~m}\) along the flat snow before coming to a stop.

Key concepts

Conservation of Energy

In physics, the principle of conservation of energy states that energy cannot be created or destroyed in an isolated system. The total energy of the system remains constant over time. This fundamental concept helps us understand a myriad of phenomena, including the movement of a snowboarder down a slope and across flat terrain.

When the snowboarder starts from rest at the top of the slope, all energy is in the form of gravitational potential energy (PE), which depends on their height above the flat snow. As they descend, this PE is converted into kinetic energy (KE), which is the energy due to motion. At the bottom of the slope, assuming no energy losses due to air resistance, all the initial PE has been converted into KE. On a flat surface, as friction exerts a force against the movement, the KE is gradually diminished until the snowboarder comes to a stop; this conversion of energy types illustrates conservation of energy in action.

Work-Energy Theorem

The work-energy theorem is another cornerstone concept in physics that relates the work done by all forces acting on a body to the change in its kinetic energy. Work is calculated as the product of the force applied and the distance over which the force is applied, in the direction of the force.

For our snowboarder, the force of kinetic friction is what does work against their motion on the flat snow. The initial KE the snowboarder has when reaching the flat ground is eventually dissipated due to the work done by friction, bringing them to a halt. According to the work-energy theorem, the work done by this frictional force must equal the loss in KE. If we note that the snowboarder has zero KE when they stop, we can infer that the work done by friction is equal to the initial KE obtained at the bottom of the slope.

Coefficient of Kinetic Friction

The coefficient of kinetic friction (μ_k) is a dimensionless scalar value that describes the ratio of the force of kinetic friction between two bodies in relative motion to the normal force pressing them together. It's a measure of how resistant a surface is to sliding motion.

In the context of our snowboarder's journey, the coefficient of kinetic friction is crucial for determining how far the snowboarder will travel on the flat snow before stopping. This value signifies how gripping or slick the interaction is between the snowboarder's board and the snow. A lower μ_k means less resistance and a farther distance before coming to rest. This coefficient, when combined with the other forces acting on the body, allows us to calculate the work done by friction, which ultimately determines the distance traveled on the flat surface.

Trigonometry in Physics

Trigonometry, the branch of mathematics dealing with the relationships of angles and sides in triangles, is frequently applied in physics to solve problems involving angles, such as the slope a snowboarder descends. Specifically, the sine function (sin) relates the angle of the slope to the height above the flat ground and the distance down the slope.

For our snowboarder scenario, we use the angle of slope (theta) and the distance traveled (d_slope) to calculate the initial height (h) using the sine function: h = d_slope × sin(theta). This height is then instrumental in determining the potential energy at the starting point, which transforms into kinetic energy that propels the snowboarder across the flat snow. Trigonometry provides the tools necessary to turn the given angle and measured slope distance into a tangible height, which is then utilized to delve into various physics calculations.