### Equations and Models for Kinetic Friction1. **Apply the Particle in Equilibrium Model in the x-Direction:** \[\sum F_x = \] \[(1) \quad = m a_x\]2. **Apply the Particle in Equilibrium Model in the y-Direction:** \[\sum F_y = \] \[(2) \quad = 0\] Use the following variables as necessary: \(g, m, n, \mu_k, \mu_s\), where the subscripts are lowercase.3. **Substitute into Equations:** Substitute \(n = mg\) from Equation (2) and \(f_k = \mu_k n\) into Equation (1). \[-\mu_k n = -\mu_k mg = m a_x\] \[a_x = \] The negative sign means the acceleration is to the **Select** in the figure. Since the velocity of the puck is to the right, the puck is slowing down. The acceleration is independent of the puck’s mass and remains constant assuming \(\mu_k\) is constant.4. **Apply the Particle under Constant Acceleration Model:** Choose the equation from the model, \[v_x^2 = v_{x0}^2 + 2a_x(x_f - x_0)\] with \(x_0 = 0\) and \(v_xf = 0\): \[0 = v_{x0}^2 + 2a_x x_f = v_{x0}^2 - 2\mu_k g x_f\]5. **Solve for the Coefficient of Kinetic Friction:** Use the variables \(g, v_{x0}, x_f\), not numerical values. \[\mu_k = \]6. **Substitute the Numerical Values:** \[\mu_k = \]**Finalize:** Note that \(\mu_k\) is dimensionless and has a low value, consistent with an object sliding on ice.### ExerciseAn experimental rocket plane lands and skids on a dry lake bed at 77.0 m/s upon touching down. Calculate the slide distance in meters, given a kinetic friction coefficient of 0.675.**Hint:**\[\_\_\_\_\_\_\_\_\_\_\_ **The Sliding Hockey Puck**A hockey puck on a frozen pond is given an initial speed of 15.0 m/s. If the puck always remains on the ice and slides 116 m before coming to rest, determine the coefficient of kinetic friction between the puck and ice.**Diagram Explanation:**The diagram shows the puck with the forces acting on it. These include:- The gravitational force, \( \vec{mg} \), acting downward.- The normal force, \( \vec{n} \), acting upward.- The force of kinetic friction, \( \vec{f_k} \), acting to the left (opposite to motion).The motion of the puck is indicated to the right.**SOLUTION****Conceptualize:** Imagine that the puck in the figure slides to the right. The kinetic friction force acts to the ---Select--- and slows the puck, which eventually comes to rest due to that force.**Categorize:** The forces acting on the puck are identified in the figure, but the text of the problem provides kinematic variables. Therefore, we categorize the problem in several ways. First, it involves modeling the puck as a particle ---Select--- in the horizontal direction: kinetic friction causes the puck to accelerate. There is no acceleration of the puck in the vertical direction, so we use the particle ---Select--- model for that direction. Furthermore, because we model the force of kinetic friction as independent of speed, the acceleration of the puck is constant. So, we can also categorize this problem by modeling the puck as a particle ---Select---.**Analyze:** First, let's find the acceleration algebraically in terms of the coefficient of kinetic friction, using Newton's second law. Once we know the acceleration of the puck and the distance it travels, the equations of kinematics can be used to find the numerical value of the coefficient of kinetic friction. The diagram in the figure shows the forces on the puck. In the diagram, how many forces are acting on the puck?- one- two- three- four(Assume that the +x-axis is to the right and the +y-axis is up along the page.)Apply the particle under a net force model in the x-direction to the puck. (Use the following as necessary: \( f_k, m, \mu_k \), and \( \mu_s \), where the subscripts are lowercase.)

The sliding Hockey Puck. The fill in the blanks. Given, Figure,

The Sliding Hockey Puck A hockey puck on a frozen pond is given an initial speed of 15.0 m/s. If the puck always remains on the ice and slides 116 m before coming to rest, determine the coefficient of kinetic friction between the puck and ice. After the puck is given an initial velocity to the right, the only external forces acting on it are the gravitational force mg, the normal force n, and the force of kinetic friction fy ni Motion mg SOLUTION Conceptualize Imagine that the puck in the figure slides to the right. The kinetic friction force acts to the ---Select--- v and slows the puck, which eventually comes to rest due to that force. Categorize The forces acting on the puck are identified in the figure, but the text of the problem provides kinematic variables. Therefore, we categorize the problem in several ways. First, it involves modeling the puck as a particle ---Select--- v in the horizontal direction: kinetic friction causes the puck to accelerate. There is no acceleration of the puck in the vertical direction, so we use the particle ---Select--- model for that direction. Furthermore, because we model the force of kinetic friction as independent of speed, the acceleration of the puck is constant. So, we can also categorize this problem by modeling the puck as a particle ---Select--- Analyze First, let's find the acceleration algebraically in terms of the coefficient of kinetic friction, using Newton's second law. Once we know the acceleration of the puck and the distance it travels, the equations of kinematics can be used to find the numerical value of the coefficient of kinetic friction. The diagram in the figure shows the forces on the puck. In the diagram, how many forces are acting on the puck? O one O two three O four (Assume that the +x-axis is to the right and the +y-axis is up along the page.) Apply the particle under a net force model in the x-direction to the puck. (Use the following as necessary: f, m, HKI and He, where the subscripts are lowercase.)

The Sliding Hockey Puck A hockey puck on a frozen pond is given an initial speed of 15.0 m/s. If the puck always remains on the ice and slides 116 m before coming to rest, determine the coefficient of kinetic friction between the puck and ice. After the puck is given an initial velocity to the right, the only external forces acting on it are the gravitational force mg, the normal force n, and the force of kinetic friction fy ni Motion mg SOLUTION Conceptualize Imagine that the puck in the figure slides to the right. The kinetic friction force acts to the ---Select--- v and slows the puck, which eventually comes to rest due to that force. Categorize The forces acting on the puck are identified in the figure, but the text of the problem provides kinematic variables. Therefore, we categorize the problem in several ways. First, it involves modeling the puck as a particle ---Select--- v in the horizontal direction: kinetic friction causes the puck to accelerate. There is no acceleration of the puck in the vertical direction, so we use the particle ---Select--- model for that direction. Furthermore, because we model the force of kinetic friction as independent of speed, the acceleration of the puck is constant. So, we can also categorize this problem by modeling the puck as a particle ---Select--- Analyze First, let's find the acceleration algebraically in terms of the coefficient of kinetic friction, using Newton's second law. Once we know the acceleration of the puck and the distance it travels, the equations of kinematics can be used to find the numerical value of the coefficient of kinetic friction. The diagram in the figure shows the forces on the puck. In the diagram, how many forces are acting on the puck? O one O two three O four (Assume that the +x-axis is to the right and the +y-axis is up along the page.) Apply the particle under a net force model in the x-direction to the puck. (Use the following as necessary: f, m, HKI and He, where the subscripts are lowercase.)