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Chapter 7: Problem 22

\(\bullet\) A 0.420 kg soccer ball is initially moving at 2.00 \(\mathrm{m} / \mathrm{s}\) . A soccer player kicks the ball, exerting a constant 40.0 \(\mathrm{N}\) force in the same direction as the ball's motion. Over what distance must her foot be in contact with the ball to increase the ball's speed to 6.00 \(\mathrm{m} / \mathrm{s} ?\)

Short Answer

Expert verified
0.168 meters.

Step by step solution

01

Determine Initial and Final Kinetic Energy

Calculate the initial kinetic energy using the formula \( KE_i = \frac{1}{2} m v_i^2 \), where \( m = 0.420 \; \text{kg} \) and \( v_i = 2.00 \; \text{m/s} \). This gives \( KE_i = \frac{1}{2} \times 0.420 \times 2.00^2 = 0.840 \; \text{J} \). Next, calculate the final kinetic energy \( KE_f = \frac{1}{2} m v_f^2 \) with \( v_f = 6.00 \; \text{m/s} \), so \( KE_f = \frac{1}{2} \times 0.420 \times 6.00^2 = 7.56 \; \text{J} \).
02

Calculate the Work Done

The work done on the ball is equal to the change in kinetic energy, \( W = KE_f - KE_i \). Substitute the values found to get \( W = 7.56 \; \text{J} - 0.840 \; \text{J} = 6.72 \; \text{J} \).
03

Relate Work Done to Force and Distance

Use the work done formula \( W = F \cdot d \), where \( F = 40.0 \; \text{N} \). Solve for \( d \) (distance): \( d = \frac{W}{F} = \frac{6.72}{40.0} = 0.168 \; \text{m} \).

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Key Concepts

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

Kinetic Energy
Kinetic energy is a fundamental concept in physics that describes the energy an object possesses due to its motion. It is calculated using the formula:
  • \( KE = \frac{1}{2} m v^2 \)
where \( m \) is the mass of the object in kilograms and \( v \) is the velocity of the object in meters per second.
The kinetic energy depends on both the mass and the square of the velocity, meaning small changes in speed can result in significant changes in kinetic energy. In our example, a soccer ball with a mass of 0.420 kg, initially moving at 2.00 m/s, has a kinetic energy of 0.840 J (joules).
When the ball's speed increases to 6.00 m/s, the kinetic energy increases to 7.56 J.
This change illustrates how the increase in velocity, even though only a factor of three, results in an increase of kinetic energy by a factor of nine, emphasizing the significance of the velocity component in the equation.
Force and Motion
Force is what causes an object to move or change its motion. In physics, when we talk about force and motion together, we're often discussing Newton's Laws of Motion. In this exercise, the soccer player exerts a force of 40.0 N on the ball, which is crucial for understanding how the ball's motion changes.
The application of force in the same direction as the motion allows the ball to accelerate, increasing its velocity from 2.00 m/s to 6.00 m/s.
  • Force \( F \) is measured in newtons (N).
  • The acceleration of the ball is due to this unbalanced force.
The player's kick provides a net force that overcomes any opposing forces, such as friction, helping the ball speed up. This process is a direct application of the work-energy theorem, where the work done by the force is converted into kinetic energy, resulting in increased speed.
Distance Calculation
To find out how far the soccer player's foot must be in contact with the ball, we use the concept of work, which in physics is the effort applied to move an object over a distance. This work is calculated by multiplying the force by the distance over which it is applied:
  • Work \( W = F \times d \)
In this problem, we've determined that the work done (the change in kinetic energy) is 6.72 J, and the force applied is 40.0 N. To find the distance \( d \), we rearrange the formula to solve for \( d \): \\[ d = \frac{W}{F} \]Substitute the values to get:
  • \( d = \frac{6.72 \, J}{40.0 \, N} = 0.168 \, m \)
This distance, 0.168 meters (or 16.8 centimeters), is the length over which the force must be applied to achieve the desired increase in speed. Understanding this calculation is key to visualizing how the application of constant force over a specific distance results in the increased kinetic energy of the soccer ball.

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Most popular questions from this chapter

\(\bullet\) \(\bullet\) A slingshot obeying Hooke's law is used to launch pebbles vertically into the air. You observe that if you pull a pebble back 20.0 \(\mathrm{cm}\) against the elastic band, the pebble goes 6.0 \(\mathrm{m}\) high. (a) Assuming that air drag is negligible, how high will the peb- ble go if you pull it back 40.0 \(\mathrm{cm}\) instead? (b) How far must you pull it back so it will reach 12.0 \(\mathrm{m}\) (c) If you pull a pebble that is twice as heavy back \(20.0 \mathrm{cm},\) how high will it go?

\(\bullet\) \(\bullet\) A spring of negligible mass has force constant \(k=\) 1600 \(\mathrm{N} / \mathrm{m}\) (a) How far must the spring be compressed for 3.20 \(\mathrm{J}\) of potential energy to be stored in it? (b) You place the spring vertically with one end on the floor. You then drop a \(1.20-\) -kg book onto it from a height of 0.80 m above the top of the spring. Find the maximum distance the spring will be compressed.

\(\bullet\) \(\bullet\) An elevator has mass \(600 \mathrm{kg},\) not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 \(\mathrm{m}\) (five floors) in \(16.0 \mathrm{s},\) and it is driven by a motor that can provide up to 40 hp to the elevator. What is the maximum number of passengers that can ride in the elevator? Assume that an average passenger has mass 65.0 \(\mathrm{kg}\) .

\(\bullet\) \(\bullet\) You and three friends stand at the corners of a square whose sides are 8.0 \(\mathrm{m}\) long in the mid- dle of the gym floor, as shown in the accompanying figure. You take your physics book and You take your physics book and push it from one person to the other. The book has a mass of \(1.5 \mathrm{kg},\) and the coefficient of kinetic friction between the book and the floor is \(\mu_{k}=0.25\) . (a) The book slides from you to Beth and then from Beth to Carlos, along the lines connect- ing these people. What is the work done by friction during this displacement? (b) You slide the book from you to Carlos along the diagonal of the square. What is the work done by friction during this displacement? (c) You slide the book to Kim who then slides it back to you. What is the total work done by fric- tion during this motion of the book? (d) Is the friction force on the book conservative or nonconservative? Explain.

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