Problem 28 - University Physics with Modern Physics

Question

A soccer ball with mass 0.420 kg is initially moving with speed 2.00 m/s. A soccer player kicks the ball, exerting a constant force of magnitude 40.0 N in the same direction as the ball's motion. Over what distance must the player's foot be in contact with the ball to increase the ball's speed to 6.00 m/s?

Step-by-Step Solution

Answer

The player's foot must be in contact with the ball for 0.168 meters.

Step 1: Apply the work-energy principle

According to the work-energy principle, the work done on an object is equal to the change in its kinetic energy. This can be expressed as: \( W = \Delta KE \). In this situation, \( W \) is the work done by the force of 40.0 N, and \( \Delta KE \) is the change in kinetic energy of the ball.

Step 2: Calculate initial and final kinetic energies

The initial kinetic energy \( KE_i \) of the ball can be calculated using the formula \( KE_i = \frac{1}{2}mv_i^2 \), where \( m = 0.420 \text{ kg} \) and \( v_i = 2.00 \text{ m/s} \). So, \( KE_i = \frac{1}{2} \times 0.420 \times (2.00)^2 = 0.84 \text{ J} \).The final kinetic energy \( KE_f \) is calculated with \( v_f = 6.00 \text{ m/s} \): \( KE_f = \frac{1}{2} \times 0.420 \times (6.00)^2 = 7.56 \text{ J} \).

Step 3: Find change in kinetic energy

The change in kinetic energy \( \Delta KE \) is \( KE_f - KE_i \). Substituting the known values: \( \Delta KE = 7.56 \text{ J} - 0.84 \text{ J} = 6.72 \text{ J} \).

Step 4: Use work done formula

The work done \( W \) by the force is equal to the product of the force and the distance over which it acts. This is expressed as \( W = F \times d \). Given \( F = 40.0 \text{ N} \) and \( W = 6.72 \text{ J} \), we can solve for \( d \) as \( d = \frac{W}{F} = \frac{6.72}{40.0} \).

Step 5: Calculate the distance

By dividing the work done by the force, \( d = \frac{6.72}{40.0} = 0.168 \text{ m} \). The player's foot must remain in contact with the ball for a distance of 0.168 meters to achieve this speed increase.

Key Concepts

Kinetic Energy

Kinetic energy describes the energy that an object possesses because of its motion. It is a fundamental concept in physics, as it determines the capacity for doing work based on an object's mass and speed. The formula to calculate kinetic energy is:\[ KE = \frac{1}{2} mv^2 \]where \( m \) is the mass and \( v \) is the speed of the object.
In the case of our soccer ball problem, we find the initial kinetic energy using its mass \( 0.420 \text{ kg} \) and initial speed \( 2.00 \text{ m/s} \). After the player's kick, the final kinetic energy is computed with the ball's new speed \( 6.00 \text{ m/s} \). Understanding these calculations is essential for applying the work-energy principle effectively.

Work Done

The concept of work done ties directly into energy transfer. In physics, work is accomplished when a force causes an object to move. Calculated as:\[ W = F \cdot d \]where \( F \) is the force applied, and \( d \) is the distance over which the force acts.
In our exercise, the work performed by the player on the soccer ball results in a change in kinetic energy. The player's force of 40.0 N increases the ball's speed via work done over a particular distance. By solving for this distance using the relationship between work and energy change, we understand that the player's kick transferred energy from the foot to the ball, evidenced by increased speed.

Force and Motion

Force is the interaction that alters an object's motion. According to Newton's second law of motion, force is connected to how an object moves or accelerates, described by:\[ F = m \cdot a \]where \( a \) is the acceleration. While this equation isn't directly used in our problem, it's crucial to understand this relationship.
The application of force by the soccer player on the ball changes the ball's velocity. The constant force of 40.0 N, applied in the direction of motion, shows how force influences motion, leading to an acceleration that increases the ball's speed from 2.00 m/s to 6.00 m/s. Understanding force allows us to see how interactions generate motion or change in speed.

Physics Problem Solving

Problem-solving in physics requires a systematic approach. Let's break down how the work-energy principle helps solve the soccer ball problem:

• Identify the given values: mass, initial and final velocities, and force.
• Compute initial and final kinetic energies to understand energy changes.
• Use relationships known from the work-energy principle to relate these energy changes to the work done.
• Set up equations with known values to solve for the unknown, in this case, the distance.

This structured approach, using physics principles, aids in solving problems by organizing information and applying relevant formulas. By breaking down the steps, students learn to approach complex scenarios with clarity and confidence.