Problem 24 - University Physics with Modern Physics

Question

A soccer ball with mass 0.420 \(\mathrm{kg}\) is initially moving with speed 2.00 \(\mathrm{m} / \mathrm{s}\) . A soccer player kicks the ball, exerting a constant force of magnitude 40.0 \(\mathrm{N}\) 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 \(\mathrm{m} / \mathrm{s} ?\)

Step-by-Step Solution

Answer

The distance is approximately 0.1365 meters.

Step 1: Identify Given Values

First, let's write down what is given in the problem. The mass of the soccer ball, \( m = 0.420 \, \mathrm{kg} \). The initial velocity of the ball, \( v_i = 2.00 \, \mathrm{m/s} \). The final velocity of the ball, \( v_f = 6.00 \, \mathrm{m/s} \). The force applied by the player, \( F = 40.0 \, \mathrm{N} \). We need to find the distance over which this force acts, \( d \).

Step 2: Use Kinetic Energy Formula

The work done by the force changes the kinetic energy of the soccer ball. Use the kinetic energy formula. The change in kinetic energy, \( \Delta KE = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 \). Substitute the given velocities and mass to find \( \Delta KE \).

Step 3: Calculate Change in Kinetic Energy

Substitute the values into the kinetic energy formula: \( \Delta KE = \frac{1}{2} \cdot 0.420 \cdot (6.00)^2 - \frac{1}{2} \cdot 0.420 \cdot (2.00)^2 \). Calculate that \( \Delta KE = 6.30 - 0.84 = 5.46 \, \mathrm{J} \).

Step 4: Use Work-Energy Principle

According to the work-energy principle, the work done on an object is equal to the change in its kinetic energy. The work done by the force is given by \( W = F \cdot d \). Therefore, \( F \cdot d = \Delta KE \).

Step 5: Solve for Distance

Now, solve for \( d \) by rearranging the equation to \( d = \frac{\Delta KE}{F} \). Substitute in \( \Delta KE = 5.46 \, \mathrm{J} \) and \( F = 40.0 \, \mathrm{N} \). Thus, \( d = \frac{5.46}{40.0} \approx 0.1365 \, \mathrm{m} \).

Step 6: Conclusion

The player's foot must be in contact with the ball over a distance of approximately 0.1365 meters to achieve the desired increase in speed.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It depends on two main factors: the mass of the object and its velocity. The formula to calculate kinetic energy is given by \( KE = \frac{1}{2} m v^2 \), where \( m \) represents mass and \( v \) is velocity. This formula highlights that kinetic energy increases with both greater mass and higher velocity.

• Mass: For a soccer ball with a mass of 0.420 kg, we can see that its mass is directly involved in calculating kinetic energy.
• Velocity: The initial velocity of 2.00 m/s and the final velocity of 6.00 m/s play crucial roles in determining the change in kinetic energy.

Understanding that kinetic energy changes as velocity changes helps us solve problems involving movements, like in our soccer ball example.

Force

Force is any interaction that, when unopposed, will change the motion of an object. It can make an object move, change direction, or stop. In physics, force is measured in Newtons (N) and is often represented by the symbol \( F \).

• In the soccer ball example, the player exerts a constant force of 40 N in the direction of the ball's motion.
• This force is responsible for increasing the ball's velocity, which in turn affects its kinetic energy.

The work-energy principle states that the work (force applied over a distance) on an object results in a change in energy, specifically kinetic energy in this context. Understanding how to calculate force and its effects is essential in solving motion-related problems.

Mass

Mass is a measure of how much matter is in an object. It’s a fundamental property that does not change regardless of location or speed. In our example, the soccer ball has a mass of 0.420 kg, which is crucial for calculating kinetic energy and understanding the effect of force.

• Mass directly affects motion; a more massive object requires more force to change its velocity than a lighter object.
• In the kinetic energy formula, mass is directly proportional to the energy, meaning more mass results in more kinetic energy.

Knowing the mass of objects in motion helps us predict how they will behave when forces are applied. This is valuable for solving physics problems and understanding everyday phenomena like sports and vehicle movements.

Velocity

Velocity is a vector quantity that refers to the speed of an object in a particular direction. Unlike speed, which only measures how fast an object is moving, velocity tells us both how fast and in which direction.

• The soccer ball in our problem starts with an initial velocity of 2.00 m/s and needs to reach a velocity of 6.00 m/s.
• The change in velocity contributes to the change in kinetic energy, as seen in the equation \( \Delta KE = \frac{1}{2} m (v_f^2 - v_i^2) \).

Understanding velocity is crucial in determining how the motion of an object changes over time. When solving problems involving motion, accurately determining initial and final velocities helps us find how forces are applied and energy exchanged.