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Chapter 4: Problem 108

A skater with an initial speed of \(7.60 \mathrm{m} / \mathrm{s}\) stops propelling himself and begins to coast across the ice, eventually coming to rest. Air resistance is negligible. (a) The coefficient of kinetic friction between the ice and the skate blades is \(0.100 .\) Find the deceleration caused by kinetic friction. (b) How far will the skater travel before coming to rest?

Short Answer

Expert verified
(a) Deceleration is 0.981 m/s虏. (b) Distance traveled is approximately 29.44 m.

Step by step solution

01

Identify Given Values

The problem states that the initial speed of the skater is \( v_0 = 7.60 \, \mathrm{m/s} \) and the coefficient of kinetic friction \( \mu_k = 0.100 \). We need to find the deceleration and the distance traveled before the skater comes to rest.
02

Calculate Deceleration Due to Friction

The force of kinetic friction can be found using \( f_k = \mu_k N \), where \( N \) is the normal force, but on a flat surface, \( N = mg \). Therefore, the frictional force is \( f_k = \mu_k mg \). Since the force of friction provides the deceleration, using Newton's second law (\( f = ma \)), we have \( ma = \mu_k mg \), hence the deceleration \( a = \mu_k g \). With \( g = 9.81 \, \mathrm{m/s^2} \), \( a = 0.100 \times 9.81 = 0.981 \, \mathrm{m/s^2} \).
03

Find Distance Using Kinematic Equation

To find the distance traveled, we use the kinematic equation \( v^2 = v_0^2 + 2a d \), where \( v = 0 \, \mathrm{m/s} \) when the skater comes to rest, \( v_0 = 7.60 \, \mathrm{m/s} \), and \( a = -0.981 \, \mathrm{m/s^2} \). Rearranging for \( d \), we get \( d = \frac{v^2 - v_0^2}{2a} \). Substituting the values, \( d = \frac{0^2 - (7.60)^2}{2(-0.981)} = \frac{-57.76}{-1.962} \approx 29.44 \, \mathrm{m} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Friction
When objects slide over each other, a resistive force called kinetic friction acts against the motion. This force depends on the nature of the surfaces in contact and the normal force. In the case of the skater, the kinetic friction force is what gradually slows them down on the ice.
  • The coefficient of kinetic friction, \( \mu_k \), is a dimensionless value that characterizes the frictional properties between two surfaces. For ice and skate blades, this was given as 0.100.
  • The normal force, \( N \), typically equals the gravitational force acting on the object when on a flat surface, calculated as \( mg \), where \( m \) is mass and \( g \, = \, 9.81 \, \mathrm{m/s^2} \) is the acceleration due to gravity.
  • The kinetic friction force \( f_k \) can thus be calculated using the equation \( f_k = \mu_k N = \mu_k mg \).
Understanding kinetic friction can be essential, as it directly determines how quickly an object loses speed when sliding on a surface.
Deceleration
Deceleration refers to the reduction in speed or the negative acceleration of an object. It occurs when the direction of the acceleration vector is opposite to the direction of motion. For the skater, the deceleration is caused by the force of kinetic friction.
The relationship between acceleration and force is given by Newton's second law, \( f = ma \). Here, the frictional force \( f_k \) is the force responsible for deceleration.
  • From the equation \( ma = f_k \), we can solve for acceleration \( a \) by dividing both sides by \( m \), yielding \( a = \frac{f_k}{m} \).
  • Since \( f_k = \mu_k mg \), we can simplify the equation to \( a = \mu_k g \).
  • In this exercise, using \( \mu_k = 0.100 \) and \( g = 9.81 \, \mathrm{m/s^2} \), the deceleration \( a \) is calculated as \( 0.981 \, \mathrm{m/s^2} \).
Understanding deceleration is crucial for determining how quickly an object can come to a stop.
Kinematic Equations
Kinematic equations describe the motion of objects under constant acceleration. They allow the calculation of various properties like velocity, acceleration, and displacement when certain initial conditions are known.
When the skater comes to a halt, we can use these equations to find the distance traveled.
  • The relevant equation in this scenario is \( v^2 = v_0^2 + 2a d \), where \( v \) is the final velocity, \( v_0 \) is the initial velocity, \( a \) is the acceleration, and \( d \) is the distance.
  • Since the skater eventually stops, \( v = 0 \), the equation simplifies to \( 0 = v_0^2 + 2a d \), which can be rearranged to solve for the distance: \( d = \frac{-v_0^2}{2a} \).
  • Substituting in the known values, \( v_0 = 7.60 \, \mathrm{m/s} \) and \( a = -0.981 \, \mathrm{m/s^2} \), the distance \( d \) is approximately 29.44 meters.
Kinematic equations are foundational tools in physics for predicting the outcomes of motion under uniform acceleration.

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