Question

A skier starts with a speed of \(2.0 \mathrm{~m} / \mathrm{s}\) and skis straight down a slope with an angle of \(15.0^{\circ}\) relative to the horizontal. The coefficient of kinetic friction between her skis and the snow is \(0.100 .\) What is her speed after 10.0 s?

Short answer

Answer: To find the skier's speed after 10.0 s, we will calculate the forces acting on her (gravitational force parallel to the slope and kinetic friction force), determine her acceleration due to these forces, and then use kinematic equations to find her speed. The final speed can be found using the equation \(v = 2.0 \mathrm{~m} / \mathrm{s} + (g\sin{\theta} - \mu g\cos{\theta})(10.0 \mathrm{s})\) with the given values for initial speed, slope angle, and coefficient of kinetic friction.

Step-by-step solution

Calculate gravitational force acting on the skier

First, we need to determine the gravitational force acting on the skier along the slope. The gravitational force can be broken into components that are parallel and perpendicular to the slope. The parallel component will be responsible for accelerating the skier down the slope. To find the gravitational force parallel to the slope, we will use the formula: \(F_{g\parallel} = mg\sin{\theta}\) Here, \(m\) is the skier's mass, \(g \approx 9.8 \mathrm{~m} / \mathrm{s^2}\) is the gravitational acceleration, and \(\theta = 15.0^{\circ}\) is the angle of the slope.

Calculate kinetic friction force acting on the skier

The kinetic friction force acting on the skier will oppose her motion down the slope. To calculate this force, we will use the formula: \(F_{f} = \mu F_{N}\) Here, \(\mu = 0.100\) is the coefficient of kinetic friction, and \(F_{N}\) is the normal force acting on the skier perpendicular to the slope. Since there is no vertical acceleration, this normal force is equal to the gravitational force component perpendicular to the slope: \(F_{N} = mg\cos{\theta}\) Thus, we get: \(F_{f} = \mu mg\cos{\theta}\)

Calculate net force and acceleration of the skier

Now, we calculate the net force acting on the skier along the slope. The net force is the difference between the gravitational force parallel to the slope and the kinetic friction force: \(F_{net} = F_{g\parallel} - F_{f}\) Using the equations we derived in steps 1 and 2: \(F_{net} = mg\sin{\theta} - \mu mg\cos{\theta}\) Since \(F_{net} = ma\), where \(a\) is the skier's acceleration, we can find the acceleration: \(a = g\sin{\theta} - \mu g\cos{\theta}\)

Use kinematic equations to find the skier's speed after 10.0 s

We are given the initial speed of the skier, \(v_{0} = 2.0 \mathrm{~m} / \mathrm{s}\), and we want to find her speed after \(t = 10.0 \mathrm{s}\). We can use the kinematic equation: \(v = v_{0} + at\) Substituting the values, we have: \(v = 2.0 \mathrm{~m} / \mathrm{s} + (g\sin{\theta} - \mu g\cos{\theta})(10.0 \mathrm{s})\) Now, we can plug in the known values of \(g\), \(\mu\), and \(\theta\) to find \(v\).

Key concepts

Gravitational Force Components

Gravitational force, also known as weight, plays a crucial role in physics, especially in kinematics problems involving motions on inclines. To fully comprehend its effect on a body sliding down a slope, like our skier, it's essential to understand its components. Gravity acts directly downward, but when an object is on an inclined surface, this force can be split into two parts: one perpendicular to the plane, and one parallel to it.

The parallel component, denoted as \(F_{g\parallel}\), is responsible for pulling the object down the incline, while the perpendicular component, \(F_{g\perp}\), interacts with the surface to produce a normal force. The normal force is significant because it affects the frictional force that opposes the downhill motion. These components are found using trigonometric functions of the incline angle, with \(F_{g\parallel} = mg\sin{\theta}\) and the normal force as \(F_{g\perp} = mg\cos{\theta}\), where \(m\) is mass, \(g\) is the acceleration due to gravity, and \(\theta\) is the angle of the incline.

Kinetic Friction

As objects move across surfaces, they experience friction - a force that resists motion. Kinetic friction, specifically, occurs when an object is sliding over a surface. In our skier's case, the kinetic friction is what opposes her motion down the snowy slope.

It is calculated using the formula \(F_{f} = \mu F_{N}\), where \(\mu\) is the coefficient of kinetic friction and \(F_{N}\) is the normal force - the force exerted by the surface perpendicular to the object. For our skier, kinetic friction can be found by multiplying the coefficient of kinetic friction with the normal force, which as we established, is the gravitational force component perpendicular to the slope \(F_{f} = \mu mg\cos{\theta}\). The important thing to remember is that kinetic friction always acts in the direction opposite to the object's motion and its magnitude is usually less than the maximum static friction.

Kinematics Equations

To predict and analyze the motion of objects, kinematics equations are employed. These equations describe the relationships between displacement, velocity, acceleration, and time without considering the forces that cause such motion. A popular equation, especially in problems involving constant acceleration like our skier's descent, is \(v = v_{0} + at\).

In this equation, \(v\) is the final velocity, \(v_{0}\) is the initial velocity, \(a\) is acceleration, and \(t\) is time. In our problem, we used this equation along with the skier's initial speed and the acceleration (caused by the net force) to determine her speed after a certain time. Kinematics equations are powerful tools in physics, allowing us to solve for various unknowns in a given motion scenario.

Coefficient of Kinetic Friction

The coefficient of kinetic friction, symbolized by \(\mu\), is a dimensionless number which characterizes the friction between two surfaces in motion. It has no units and is specific to the pair of surfaces in contact; for instance, the skier's skis on snow. The value of \(\mu\) helps us determine the frictional force by multiplying it with the normal force exerted by a surface.

In our exercise, the coefficient of kinetic friction plays a vital role in calculating the skier's net force and subsequent acceleration. Having a lower coefficient means less resistance and faster acceleration, while a higher coefficient means the opposite. Understanding this coefficient is key to predicting how much an object will slow down or speed up due to friction when moving on a surface.