Question

A softball, of mass \(m=0.250 \mathrm{~kg}\), is pitched at a speed \(v_{0}=26.4 \mathrm{~m} / \mathrm{s}\). Due to air resistance, by the time it reaches home plate it has slowed by \(10.0 \% .\) The distance between the plate and the pitcher is \(d=15.0 \mathrm{~m}\). Calculate the average force of air resistance, \(F_{\text {air }}\) that is exerted on the ball during its movement from the pitcher to the plate.

Short answer

Answer: The average force of air resistance on the ball during its travel is approximately \(F_{air} \approx -1.98 \text{ N}\) (negative sign indicates the force is opposing the motion).

Step-by-step solution

Calculate the final speed of the softball

The initial speed, \(v_0\), of the softball is 26.4 m/s and the air resistance reduces its speed by 10%. We can calculate the final speed, \(v_f\) as follows: \(v_f = v_0 - (\text{reduction percentage}) \times v_0\) \(v_f = 26.4 - 0.1 \times 26.4\) \(v_f = 26.4 - 2.64 = 23.76 \text{ m/s}\)

Calculate initial and final kinetic energies

The initial and final kinetic energies of the softball can be calculated using the following formula: \(KE = \frac{1}{2}mv^2\) Initial kinetic energy, \(KE_i\): \(KE_i = \frac{1}{2} (0.250)(26.4)^2\) Final kinetic energy, \(KE_f\): \(KE_f = \frac{1}{2} (0.250)(23.76)^2\)

Use the work-energy principle to find the work done by air resistance

According to the work-energy principle, the work done by air resistance is equal to the change in kinetic energy. \(W_{air} = KE_f - KE_i\) Calculate the work done by air resistance: \(W_{air} = \frac{1}{2}(0.250)(23.76)^2 - \frac{1}{2}(0.250)(26.4)^2\)

Calculate the average force of air resistance

The force of air resistance can be calculated using the formula for work, \(F_{air}=\frac{W_{air}}{d}\), where \(d\) is the distance between the pitcher and the plate. \(F_{air} = \frac{W_{air}}{15.0 \text{ m}}\) Using the calculated value of \(W_{air}\), calculate the average force of air resistance: \(F_{air} = \frac{ \frac{1}{2}(0.250)(23.76)^2 - \frac{1}{2}(0.250)(26.4)^2 }{15.0}\) Now, simply compute the value for \(F_{air}\).

Key concepts

Kinetic Energy

Kinetic energy is the energy an object has because of its motion. It's a key concept in physics, especially when analyzing the motion of objects like a softball. The formula to calculate kinetic energy (KE) is: \[ KE = \frac{1}{2} mv^2 \] where \( m \) is the mass of the object and \( v \) is its velocity. For our softball problem, we compute both the initial and final kinetic energies. When the softball is pitched, it has an initial speed (\( v_0 \text{=} 26.4 \, \text{m/s} \)), and at this moment, you can calculate its initial kinetic energy, \( KE_i \). As it travels and experiences air resistance, its speed decreases by 10%, resulting in a new speed of \( 23.76 \, \text{m/s} \). This new speed allows us to find the final kinetic energy, \( KE_f \). Changes in kinetic energy help us understand how air resistance affects the motion of the softball.

Work-Energy Principle

The work-energy principle is a vital concept that connects the work done on an object and its kinetic energy change. According to this principle, the work done by all forces acting on an object is equal to the change in its kinetic energy. This can be expressed as: \[ W = KE_f - KE_i \] where \( W \) is the work done, \( KE_f \) is the final kinetic energy, and \( KE_i \) is the initial kinetic energy.
When we have a force acting over a distance, like air resistance on a softball, it does work on the ball. The work done is precisely that loss in kinetic energy due to the slowing effect. To solve for the work done by air resistance in this problem, we use the computed \( KE_i \) and \( KE_f \) values and find the difference.

Air Resistance

Air resistance plays a crucial role in many physics problems involving moving objects. It is a force that opposes motion through the air, also known as drag. For the softball, air resistance slows down its velocity as it travels from the pitcher to the batter.
It is essential to understand that air resistance depends on several factors, like the object's speed, size, and shape. The faster and larger the object, often the greater the air resistance. In our softball problem, air resistance is the key factor causing the softball to reduce its speed by 10% over a distance of 15 meters. This slowing effect is not just a numbers game, but a real world influence of forces in motion.

Average Force

The concept of average force is used to simplify the understanding of force interactions that happen over a period of time or distance. When an object is subject to forces, like air resistance when moving from one point to another, it's convenient to talk about an average force rather than calculating force at every instant.
To find the average force of air resistance acting on the softball, we apply the formula: \[ F_{\text{air}} = \frac{W_{\text{air}}}{d} \] where \( W_{\text{air}} \) is the work done by air resistance, and \( d \) is the distance over which the force acts, in this case, 15 meters.
This average force provides a useful approximation to understand the effect of air resistance over the journey of the softball from pitcher to plate.

Softball Physics

Studying the physics of a softball pitch helps us understand how forces and motions interact in a real-world scenario. Several factors come into play: the initial pitch speed, the mass of the softball, the effects of air resistance, and the distance traveled.
In this problem, we're tasked with calculating the average force exerted by air resistance, which causes a decrease in speed as the ball travels to the batter. By grasping these concepts, students can appreciate how physics applies not just in classrooms, but also in playing fields.

• The initial and final speeds tell us how fast the ball travels.
• The work-energy principle lays out how energy changes.
• Air resistance reveals how forces oppose motion.

This isn't just theory but a practical exploration of how physics governs the world around us.