Problem 26 - Physics for Scientists and Engineers with Modern Physics

Question

(II) \(\mathrm{A}\) 75-kg snowboarder has an initial velocity of 5.0 \(\mathrm{m} / \mathrm{s}\) at the top of a \(28^{\circ}\) incline (Fig. \(36 ) .\) After sliding down the 110 -m long incline (on which the coefficient of kinetic friction is \(\mu_{k}=0.18\) ), the snowboarder has attained a velocity \(v .\) The snowboarder then slides along a flat surface (on which \(\mu_{k}=0.15\) and comes to rest after a distance \(x .\) Use Newton's second law to find the snowboarder's acceleration while on the incline and while on the flat surface. Then use these accelerations to determine \(x .\)

Step-by-Step Solution

Answer

The snowboarder's acceleration on the incline is approximately \(2.81 \, \mathrm{m/s^2}\). On the flat surface, the acceleration is \(1.47 \, \mathrm{m/s^2}\), and the stopping distance \(x\) is about \(36 \, \mathrm{m}\).

Step 1: Analyze Forces on the Incline

First, identify the forces acting on the snowboarder while sliding down the incline. These include gravitational force, normal force, and frictional force. The gravitational force parallel to the incline is given by \( F_{g,\parallel} = mg\sin(\theta) \), where \( m = 75 \, \mathrm{kg} \) and \( \theta = 28^{\circ} \). The normal force is \( F_N = mg\cos(\theta) \). The frictional force is \( F_{\text{friction}} = \mu_k \cdot F_N = \mu_k \cdot mg\cos(\theta) \).

Step 2: Calculate Net Force on the Incline

The net force acting along the incline is \( F_{\text{net}} = F_{g,\parallel} - F_{\text{friction}} = mg\sin(\theta) - \mu_k mg\cos(\theta) \). Substitute the known values to find \( F_{\text{net}} \).

Step 3: Determine Acceleration on the Incline

Using Newton's second law, \( F_{\text{net}} = ma \), solve for the acceleration \( a_{\text{incline}} \) by dividing the net force by the mass: \( a_{\text{incline}} = g(\sin(\theta) - \mu_k \cos(\theta)) \). Input the known values to find numerical acceleration.

Step 4: Calculate Final Velocity on the Incline

Use the kinematic equation \( v^2 = u^2 + 2a_{\text{incline}}d \) to find the final velocity \( v \) at the bottom of the incline, where \( u = 5.0 \, \mathrm{m/s} \) and \( d = 110 \, \mathrm{m} \). Solve for \( v \).

Step 5: Analyze Forces on the Flat Surface

Once on the flat surface, the only forces are the normal force and kinetic friction. The frictional force is \( F_{\text{friction,flat}} = \mu_k mg \), where \( \mu_k = 0.15 \).

Step 6: Calculate Acceleration on the Flat Surface

Apply Newton's second law: \( F_{\text{net,flat}} = \mu_k mg = ma_{\text{flat}} \). Solve for \( a_{\text{flat}} \) to find the acceleration on the flat surface: \( a_{\text{flat}} = \mu_k g \).

Step 7: Determine Distance to Stop on Flat Surface

Using the kinematic equation \( v^2 = u^2 + 2a_{\text{flat}}x \) and knowing final velocity \( v = 0 \), solve for the distance \( x \). Rearrange to get \( x = -\frac{u^2}{2a_{\text{flat}}} \) and substitute the known velocity and acceleration.

Key Concepts

Kinetic Friction

Kinetic friction is the force resisting the motion of an object sliding on a surface. It depends on two main factors: the coefficient of kinetic friction between the two surfaces and the normal force exerted by the object. For the snowboarder on the incline, kinetic friction acts in the opposite direction of motion.When the snowboarder slides down the

incline

, the kinetic friction force is calculated by the formula: \[ F_{\text{friction}} = \mu_k \cdot F_N = \mu_k \cdot mg\cos(\theta) \] where \( \mu_k = 0.18 \) is the coefficient of kinetic friction, \( m \) is the mass of the snowboarder (75 kg), \( g \) is the acceleration due to gravity (approximately 9.8 m/s²), and \( \theta \) is the angle of the incline (28°).On the flat surface, the coefficient of kinetic friction changes to \( \mu_k = 0.15 \), affecting how quickly the snowboarder decelerates. Here, the frictional force becomes:\[ F_{\text{friction,flat}} = \mu_k mg \] This dynamic highlights how kinetic friction can vary based on surface and slope, impacting the snowboarder's motion.

Acceleration Calculation

Calculating acceleration is crucial for understanding how the snowboarder moves on both the incline and flat surfaces. By applying Newton’s Second Law, \( F = ma \), we can compute acceleration by dividing the net force by the snowboarder's mass.On the incline, after determining the net force: \[ F_{\text{net}} = mg\sin(\theta) - \mu_k mg\cos(\theta) \] We find acceleration with:\[ a_{\text{incline}} = g(\sin(\theta) - \mu_k \cos(\theta)) \] Plug in the values: \( g = 9.8 \ \text{m/s}^2 \), \( \theta = 28° \), and \( \mu_k = 0.18 \) to compute the snowboarder's acceleration on the incline.
On the flat surface, only friction opposes motion because the slope angle \( \theta \) is now zero. Using the formula:\[ a_{\text{flat}} = \mu_k g \] With \( \mu_k = 0.15 \), calculate the specific acceleration eliminating the need to consider any incline angle. Understanding these calculations enables predicting and analyzing motion in real scenarios.

Inclined Plane

Inclined planes are surfaces tilted at an angle relative to the horizontal. They help in simplifying complex force and motion components into more manageable calculations.
On an inclined plane, the gravitational force acting on an object can be split into two components:

• Parallel component pushing the snowboarder down the slope: \( F_{g,\parallel} = mg\sin(\theta) \)
• Perpendicular component exerted as a normal force: \( F_N = mg\cos(\theta) \)

The snowboarder's weight and the angle of the incline determine how fast the snowboarder will accelerate down the slope.
Knowing how to break down forces on an inclined plane makes it easier to evaluate each force's contribution. In this problem, understanding inclined planes allows predictions about speed buildup and helps determine other factors like travel distances and deceleration on flat surfaces, which ultimately enables better decision-making and insights into real-world mechanics.