Problem 29 - College Physics

Question

A hockey puck leaves a player's stick with a speed of \(9.9 \mathrm{~m} / \mathrm{s}\) and slides \(32.0 \mathrm{~m}\) before coming to rest. Find the coefficient of friction between the puck and the ice.

Step-by-Step Solution

Answer

The coefficient of friction is approximately 0.156.

Step 1: Identify the variables

We know the initial speed of the hockey puck is \( v_i = 9.9 \, \text{m/s} \), the final speed is \( v_f = 0 \, \text{m/s} \) (since it comes to rest), and the distance travelled is \( d = 32.0 \, \text{m} \). We need to find the coefficient of friction, \( \mu \).

Step 2: Apply the kinematic equation

Use the kinematic equation \( v_f^2 = v_i^2 + 2a d \). Substituting the known values gives \( 0 = (9.9 \, \text{m/s})^2 + 2a (32.0 \, \text{m}) \).

Step 3: Rearrange to find acceleration

By rearranging the kinematic equation for acceleration, we get \( a = -\frac{(9.9 \, \text{m/s})^2}{2 \times 32.0 \, \text{m}} \). Calculate \( a \) to get \( a \approx -1.53 \, \text{m/s}^2 \).

Step 4: Relate friction to acceleration

The acceleration due to friction is related to the frictional force by \( a = \mu g \) where \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity. Thus, \( \mu = \frac{a}{g} \).

Step 5: Solve for the coefficient of friction

Substitute \( a \approx -1.53 \, \text{m/s}^2 \) and \( g = 9.81 \, \text{m/s}^2 \) into \( \mu = \frac{-1.53}{9.81} \) to find \( \mu \approx 0.156 \).

Key Concepts

Kinematic Equations

Kinematic equations are fundamental tools in physics that help us describe the motion of objects. They relate the variables such as displacement, velocity, acceleration, and time. In the hockey puck problem, the kinematic equation used is \( v_f^2 = v_i^2 + 2ad \). This equation helps determine how the puck's speed changes as it slides across the ice. Here:

• \( v_f \) is the final velocity, which is \( 0 \) m/s since the puck comes to a rest.
• \( v_i \) is the initial velocity, given as \( 9.9 \) m/s.
• \( a \) is the acceleration, which we'll find using this equation.
• \( d \) is the displacement, \( 32.0 \) m in this case.

By rearranging the equation, \( a \) can be solved, allowing us to understand how the puck slows down. Understanding kinematic equations is vital in solving motion problems, as they provide a direct link between the motion parameters.

Acceleration due to Gravity

The acceleration due to gravity, denoted by \( g \), is an essential constant in physics, approximately \( 9.81 \, \text{m/s}^2 \) on Earth's surface. This value represents the acceleration objects experience in free fall, assuming only gravity is acting on them. In the context of the hockey puck problem, we're dealing with horizontal motion on a frictional surface. Gravity impacts the normal force which is perpendicular to the surface, influencing the frictional force between two surfaces.

• The frictional force is pivotal here, as it is what slows down the puck.
• It is calculated using \( \mu g \), where \( \mu \) is the coefficient of friction.

This frictional force is what provides the deceleration to the puck, and \( g \) helps in quantifying that interaction. With the grip of gravity, it becomes possible to relate acceleration to friction in real-world problems.

Physics Problem-Solving

Physics problem-solving often begins with identifying known values and required solutions. This systematic approach ensures clarity. In our exercise, identifying the initial velocity, final velocity, and displacement paved the path for determining the coefficient of friction. Here's how effective problem-solving is structured:

• Understand the Problem: Know what you are given and what you need to find. Write down all known variables.
• Choose the Right Equations: For motion, kinematic equations are often suitable. Select the one connecting all known variables.
• Rearrange and Solve: Manipulate equations algebraically to isolate the unknown you need to find. This extraction was crucial here to find acceleration and later, friction.
• Interpret the Results: Ensure your findings make sense in the real-world context. A coefficient of friction value should be reasonable for ice.

With patience, physics problem-solving becomes a logical deduction, applying equations to gain insights from the physical world. Make sure to practice breaking down complex problems into smaller, manageable parts. This will significantly assist in developing a robust problem-solving skill set.