Question

A ski jumper glides down a \(30.0^{\circ}\) slope for \(80.0 \mathrm{ft}\) before taking off from a negligibly short horizontal ramp. If the jumper's takeoff speed is \(45.0 \mathrm{ft} / \mathrm{s}\), what is the coefficient of kinetic friction between skis and slope? Would the value of the coefficient of friction be different if expressed in SI units? If yes, by how much would it differ?

Short answer

Answer: The coefficient of kinetic friction between the skis and the slope can be determined using work-energy theorem and equations of motion. Its value doesn't change when expressed in SI units as it is a dimensionless quantity.

Step-by-step solution

Calculate the change in gravitational potential energy of the ski jumper

To solve for the coefficient of kinetic friction, we first need to find out the change in gravitational potential energy. We can do this using the formula \(\Delta{PE} = mgh\), where \(m\) is the mass of the ski jumper, \(g\) is the acceleration due to gravity (\(32.2 \mathrm{ft/s^2}\)), and \(h\) is the vertical height. We can find the height by solving \(h = 80.0\mathrm{ft} \times \sin{30.0^{\circ}}\).

Calculate the change in kinetic energy of the ski jumper

The change in kinetic energy, \(\Delta{KE}\), can be found using the formula \(\Delta{KE} = \frac{1}{2}m(v_f^2 - v_i^2)\), where \(v_f\) is the final velocity (\(45.0 \mathrm{ft/s}\)) and \(v_i\) is the initial velocity (assumed to be 0). This simplifies to \(\Delta{KE} = \frac{1}{2}m(45.0^2 - 0)\).

Calculate the work done against friction

The work-energy theorem states that the work done against friction equals the change in kinetic energy minus the change in potential energy, i.e., \(W = \Delta{KE} - \Delta{PE}\). Plugging in the values calculated in Steps 1 and 2, we get \(W = \frac{1}{2}m(45.0^2 - 0) - mgh\).

Determine the coefficient of kinetic friction

We can find the coefficient of kinetic friction, \(\mu_k\), using the formula \(W = \mu_k \times N \times d\), where \(N\) is the normal force and \(d\) is the distance travelled down the slope. Since the ski jumper is moving along the slope, the normal force is equal to \(mg\cos{30.0^{\circ}}\). Substituting this into the equation, we get \(\frac{1}{2}m(45.0^2 - 0) - mgh = \mu_k \times mg\cos{30.0^{\circ}} \times 80.0\mathrm{ft}\). As the mass cancels out, we can solve for \(\mu_k\).

Check if the value of the coefficient of friction is different in SI units

The coefficient of kinetic friction is a dimensionless quantity. Therefore, its value doesn't change regardless of the unit system used. So, we have determined the coefficient of kinetic friction between the skis and the slope, and we've confirmed that this value would not be different when expressed in SI units.

Key concepts

Kinetic Friction

Kinetic friction is a force that opposes the relative motion between two surfaces that are sliding past each other. It is an important concept in physics because it affects how objects move across a surface. In the context of our problem, kinetic friction is the force working against the ski jumper's motion as they slide down the slope. Understanding this concept involves recognizing:

• **Coefficient of Kinetic Friction ( \( \mu_k \)):** A dimensionless number that represents the ratio of the force of kinetic friction to the normal force acting between the two surfaces.
• **Normal Force ( \( N \)):** This is the perpendicular force exerted by a surface on an object resting on it. For a slope, it's calculated as \( mg \cos{\theta} \), where \( \theta \) is the angle of the incline.

In our ski jumper scenario, \( \mu_k \) is calculated using the relationship provided by the work-energy theorem, where you equate the work done by friction to the change in energy. The coefficient of kinetic friction calculated from this setup reflects all the forces the ski jumper encounters on the slope.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy stored in an object as a result of its height above the ground. The higher an object is, the more GPE it possesses because it can fall further. Calculating gravitational potential energy can be done using the equation: \[ \Delta{PE} = mgh \]where:

• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity,
• \( h \) is the height above a reference level.

For the ski jumper, as they glide down the slope, their height decreases, decreasing their gravitational potential energy. This decrease in GPE is what contributes to their gain in speed and kinetic energy. Understanding this relationship helps in analyzing energy transformations in physics, particularly in how potential energy is converted into kinetic energy.

Work-Energy Theorem

The work-energy theorem is a fundamental concept in physics that describes the relationship between work done and the change in kinetic energy of an object. It states that the work done by all the forces acting on an object equates to the change in its kinetic energy. In mathematical terms:\[ W = \Delta{KE} = \frac{1}{2}m(v_f^2 - v_i^2) \]The beauty of this theorem lies in how it connects work, a concept involving force and distance, to kinetic energy—a form of energy due to motion. For the ski jumper:

• **Initial and Final Velocities ( \( v_i \) and \( v_f \)):** The jumper begins from rest, simplifying \( v_i \) to zero, and moves to \( v_f \), the speed at takeoff.
• **Work Done by Friction:** It's calculated by taking the change in kinetic energy and subtracting the loss in gravitational potential energy.

By understanding the work-energy theorem, students can apply it to a variety of real-world scenarios, such as calculating frictional effects or determining energy transfer in systems like the ski descent.