Question

A skier starts with a speed of \(2.00 \mathrm{~m} / \mathrm{s}\) 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: The speed of the skier after 10 seconds is approximately 17.38 meters per second.

Step-by-step solution

Determine the gravitational force acting on the skier

The first step is to find the gravitational force acting on the skier along the slope. This can be found by: \(f_g = m \cdot g \cdot sin(\theta)\) where \(f_g\) is the gravitational force along the slope, \(m\) is the mass of the skier, \(g\) is the acceleration due to gravity, and \(\theta\) is the angle of the slope. Don't worry about the mass of the skier because it'll cancel out later when we calculate the acceleration of the skier.

Calculate the frictional force acting on the skier

The frictional force acting on the skier is given by: \(f_f = \mu \cdot f_n\) Since the skier is moving along the angled slope, the normal force is equal to the component of gravitational force acting perpendicular to the slope: \(f_n = m \cdot g \cdot cos(\theta)\) So, the frictional force is: \(f_f = \mu \cdot m \cdot g \cdot cos(\theta)\)

Calculate the net force acting on the skier

Now we'll determine the net force acting on the skier, which is given by: \(f_{net} = f_g - f_f\) Substitute the expressions for \(f_g\) and \(f_f\) to get: \(f_{net} = m \cdot g \cdot sin(\theta) - \mu \cdot m \cdot g \cdot cos(\theta)\)

Determine the acceleration of the skier

To find the acceleration of the skier, we'll use Newton's second law: \(a = \frac{f_{net}}{m}\) Plug in the expression for the net force: \(a = \frac{m \cdot g \cdot sin(\theta) - \mu \cdot m \cdot g \cdot cos(\theta)}{m}\) The mass of the skier cancels out, so we get: \(a = g \cdot (sin(\theta) - \mu \cdot cos(\theta))\) Now, plug in the values of \(\theta = 15^{\circ}\), \(\mu = 0.100\), and \(g = 9.81 \mathrm{~m} / \mathrm{s}^2\): \(a = 9.81 \cdot ( sin(15) - 0.100 \cdot cos(15))\) Calculate the acceleration: \(a \approx 1.538 \mathrm{~m}/\mathrm{s}^2\)

Calculate the final speed of the skier

Finally, use the kinematic equation to find the final velocity after 10 seconds: \(v_f = v_i + a \cdot t\) Plug in the initial velocity (\(v_i = 2.00 \mathrm{~m}/\mathrm{s}\)), the acceleration (\(a \approx 1.538 \mathrm{~m}/\mathrm{s}^2\)), and the time (\(t = 10.0 \mathrm{~s}\)): \(v_f = 2.00 \mathrm{~m}/\mathrm{s} + 1.538 \mathrm{~m}/\mathrm{s}^2 \cdot 10.0 \mathrm{~s}\) Calculate the final speed: \(v_f \approx 17.38 \mathrm{~m}/\mathrm{s}\) The skier's speed after 10 seconds is approximately 17.38 meters per second.

Key concepts

Gravitational Force

Gravitational force is a fundamental interaction that attracts two bodies towards each other based on their masses and the distance between them. In physics problems, it's often represented in terms of the weight force a mass feels due to gravity. When an object is on an incline, like the skier in our exercise on a slope, the force of gravity can be broken down into two components: one acting perpendicular to the surface, and one acting parallel along the slope.

The component acting along the surface is what causes the object to slide down the slope and can be calculated using the formula: \(f_g = m \cdot g \cdot \sin(\theta)\), where \(\theta\) is the angle of the incline. This understanding is crucial, as it plays a key role in determining the acceleration of the object down the slope, which leads to the calculation of speed or distance given certain conditions.

Frictional Force

Frictional force is the resistive force that acts opposite to the motion (or intended motion) of two surfaces in contact. It's a key concept in physics because it accounts for the resistance an object experiences as it moves across a surface. The magnitude of the frictional force is generally calculated using the coefficient of friction and the normal force, through the equation \(f_f = \mu \cdot f_n\).

In our skier's problem, the frictional force acts in the opposite direction of her downhill motion, effectively reducing her net acceleration down the slope. It is proportional to the strength of the normal force – the component of gravitational force perpendicular to the surface, determined by \(f_n = m \cdot g \cdot \cos(\theta)\) – and the coefficient of kinetic friction, which is a measure of how slippery or grippy the surface is.

Kinematic Equations

Kinematic equations are a set of formulas that describe the motion of objects without considering the forces that cause the motion. These equations are essential tools for solving problems in mechanics, especially when it comes to analyzing objects moving with uniform acceleration.

One key kinematic equation is \(v_f = v_i + a \cdot t\), which relates the final velocity (\(v_f\)) of an object to its initial velocity (\(v_i\)), the acceleration (\(a\)), and the time (\(t\)) over which the acceleration occurs. In the context of our skier, the kinematic equation is used to calculate her speed after a certain time, giving a clear, final numerical result of her velocity down the slope.

Newton's Second Law

Newton's second law of motion is one of the most fundamental concepts in classical physics. It states that the acceleration (\(a\)) of an object is directly proportional to the net force (\(f_{net}\)) acting upon it and inversely proportional to its mass (\(m\)), which can be formulated as \(a = \frac{f_{net}}{m}\).

This law explains how the forces acting on the skier – gravitational pull down the slope and the frictional force going against the motion – work together to determine her acceleration. The net force is the difference between these two forces, and since the mass of the skier cancels out in our exercise, the acceleration down the slope can be found easily by subtracting the two, leading to the calculation of her final speed through the kinematic equations.