Problem 71 - College Physics

Question

A block is projected with a speed of \(3.0 \mathrm{~m} / \mathrm{s}\) on a horizontal surface. If the coefficient of kinetic friction between the block and the surface is 0.60 , how far does the block slide before coming to rest?

Step-by-Step Solution

Answer

The block slides approximately 0.77 meters before stopping.

Step 1: Identify Known Values

We are given that the initial velocity \( v_0 = 3.0 \, \text{m/s} \), the coefficient of kinetic friction \( \mu_k = 0.60 \), and the final velocity \( v = 0 \) (since the block comes to rest).

Step 2: Determine Force of Friction

The force of kinetic friction \( f_k \) can be calculated using the formula: \( f_k = \mu_k F_n \), where \( F_n \) is the normal force. On a horizontal surface, \( F_n = mg \), where \( m \) is the mass of the block, and \( g \) is the acceleration due to gravity (\( g = 9.8 \, \text{m/s}^2 \)). Thus, \( f_k = \mu_k mg \).

Step 3: Apply Work-Energy Principle

The work done by the friction force is equal to the change in kinetic energy. The initial kinetic energy is \( \frac{1}{2} mv_0^2 \) and the final kinetic energy is 0. Thus, the work done, \( W = f_k \cdot d = \frac{1}{2} mv_0^2 \), where \( d \) is the distance the block slides.

Step 4: Solve for Distance

Equate the work done to the change in kinetic energy: \( \mu_k mgd = \frac{1}{2} mv_0^2 \). Solve for \( d \):\[ d = \frac{\frac{1}{2} v_0^2}{\mu_k g} = \frac{\frac{1}{2} (3.0)^2}{0.60 \times 9.8} \approx \frac{4.5}{5.88} \approx 0.765 \. \text{meters} \].

Key Concepts

Understanding the Work-Energy Principle

The work-energy principle is a fundamental concept in physics that links the work done by forces on an object to its change in kinetic energy. In simple terms, it states that the work performed by all forces acting on an object is equal to the change in the object's kinetic energy. This principle allows us to solve problems involving the motion of objects when forces like friction are at play.

To apply the work-energy principle:

• Calculate how much work is done by all forces acting on the object.
• Assess how this work affects the object's kinetic energy.
• Use changes in kinetic energy to find unknowns like distance, speed, or stopping force.

In the context of our problem, the work done by friction is what brings the block to a halt. By calculating this, we can determine how far the block slides.

Exploring Kinetic Energy

Kinetic energy refers to the energy an object possesses due to its motion. When something is moving, it carries energy that can be transferred to other objects or converted to other forms. For an object of mass \( m \) and velocity \( v \), its kinetic energy \( KE \) is given by the equation: \( KE = \frac{1}{2} mv^2 \).

In our exercise, the initial kinetic energy of the block was determined by its speed and mass. Since the block eventually stops, its final kinetic energy is zero. This loss in kinetic energy is exactly matched by the work done against the force of friction.
Understanding kinetic energy helps us see how motion is intertwined with energy changes, enabling us to solve for the distance the block travels until it stops.

Mindful Physics Problem-Solving

Physics problem-solving often involves identifying what you know and what you need to find. Here's a concise strategy:

• Identify given variables, such as initial speed or coefficients of friction.
• Recognize what principles apply, like the work-energy principle for stopping distances.
• Relate the variables using mathematical formulas relevant to the physics concepts involved.

In the block example, recognizing the link between work done by friction and the change in kinetic energy is crucial. By setting the work (force times distance) equal to the lost kinetic energy, you can solve for unknowns like the sliding distance. Understand the problem deeply, break it into manageable steps, and you'll navigate physics challenges with confidence.