Problem 68 - College Physics

Question

A hockey player hits a puck with his stick, giving the puck an initial speed of \(5.0 \mathrm{~m} / \mathrm{s}\). If the puck slows uniformly and comes to rest in a distance of \(20 \mathrm{~m}\), what is the coefficient of kinetic friction between the ice and the puck?

Step-by-Step Solution

Answer

The coefficient of kinetic friction is approximately 0.064.

Step 1: Understand the Known Values

We know the initial velocity of the puck \(v_i = 5.0 \mathrm{~m/s}\) and the final velocity \(v_f = 0 \mathrm{~m/s}\) since the puck comes to rest. The distance the puck travels before stopping is \(d = 20 \mathrm{~m}\). We need to find the coefficient of kinetic friction \(\mu_k\).

Step 2: Use the Equation of Motion to Find Acceleration

The equation of motion that relates initial velocity, final velocity, acceleration, and distance is given by:\[ v_f^2 = v_i^2 + 2a d \]Plugging in the known values, we solve for \(a\):\[ 0 = (5.0)^2 + 2a(20) \]\[ 0 = 25 + 40a \]\[ 40a = -25 \]\[ a = -\frac{25}{40} = -0.625 \mathrm{~m/s^2} \]

Step 3: Relate Acceleration to Friction

Using Newton's second law, the frictional force \(F_f\) is given by \(F_f = ma\), where \(m\) is the mass of the puck. However, since we are looking for the coefficient of kinetic friction, we can use \(F_f = \mu_k N\), where \(N\) is the normal force, \(N = mg\). Thus:\[ \mu_k mg = ma \]\[ \mu_k g = |a| \]Since the motion is horizontal, and assuming a frictionless surface, we divide acceleration by gravity \(g = 9.8 \mathrm{~m/s^2}\) to find \(\mu_k\):\[ \mu_k = \frac{|a|}{g} = \frac{0.625}{9.8} \]

Step 4: Calculate the Coefficient of Kinetic Friction

Substitute the known values to find \(\mu_k\):\[ \mu_k = \frac{0.625}{9.8} = 0.064 \]

Key Concepts

Equation of Motion

The equation of motion is a mathematical tool that helps us link various elements of motion, such as velocity, acceleration, and distance. This particular equation to find the acceleration when the puck is in motion is written as:

• \[ v_f^2 = v_i^2 + 2a d \]

By using this equation, we can understand how an object moves over time. It shows the relationship between the initial velocity \(v_i\), the final velocity \(v_f\), the acceleration \(a\), and the distance covered \(d\). By substitution, we can solve for the unknown parameter, such as acceleration in this case. If you rearrange the equation to solve for \(a\), you get:

• \[ a = \frac{v_f^2 - v_i^2}{2d} \]

Plugging in the known values is straightforward. This helps to determine how quickly an object's velocity changes with respect to time when some forces act upon it.

Newton's Second Law

Newton's second law is a fundamental principle of physics that relates to how forces affect the motion of objects. It can be expressed through the equation:

• \[ F = ma \]

This law tells us that force is the product of an object's mass \(m\) and its acceleration \(a\). In the context of our problem, this principle is used to relate the frictional force experienced by the puck to its mass and acceleration. Newton's second law explains how much force is necessary to change an object's velocity, helping us calculate the force required to make the puck stop.
When applied to the hockey puck, the force due to friction that opposes its motion can also be described in terms of kinetic friction as:

• \[ F_f = \mu_k N \]

where \( \mu_k \) is the coefficient of kinetic friction and \(N\) is the normal force, provided by the puck's weight.

Acceleration

Acceleration is a measure of how fast an object's velocity changes over time. It is a vector quantity, meaning it has both magnitude and direction. The puck in this problem decelerates uniformly, reducing its speed to zero. The calculation of acceleration involves understanding how quickly the puck comes to rest as it slides over the ice.

• From the equation of motion, we determine that:\[ a = -0.625 \text{ m/s}^2 \]

The negative sign indicates that the puck is slowing down. Understanding acceleration helps us to analyze whether an object is speeding up, slowing down, or changing direction during its motion. In practical terms, knowing the acceleration is crucial in calculating other dynamics like the distance required to stop an object, or the force exerted during its motion.

Coefficient of Friction

The coefficient of friction is a unitless value that represents the ratio of the force of friction between two bodies and the force pressing them together. In our scenario, it tells us how much frictional resistance the puck experiences as it glides across the ice. The coefficient of kinetic friction \( \mu_k \) can be calculated using the formula:

• \[ \mu_k = \frac{|a|}{g} \]

where \(g\) is the acceleration due to gravity, approximately \(9.8 \text{ m/s}^2\). In the case of the puck, using the calculated acceleration, we find:

• \[ \mu_k = \frac{0.625}{9.8} \approx 0.064 \]

This tells us how grippy or slippery the ice is in relation to the puck. A lower value of \(\mu_k\) indicates less friction, hence a smoother sliding surface. Understanding the coefficient of friction aids in designing surfaces and materials for specific needs, whether for maximizing speed or for providing stability and control.