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 exerted on the softball is approximately 1.367 N.

Step-by-step solution

Determine final velocity v

Since the softball slows down by 10%, the final velocity, v, is 90% of the initial velocity, \(v_0\). Calculate the final velocity as follows: \(v = 0.90 \times v_0\) \(v = 0.90 \times 26.4 m/s\) \(v = 23.76 m/s\)

Determine the time t

Since the velocity changes linearly with time, we can take the average velocity during the motion and use it to find the time it takes to travel the distance. The average velocity is given by: \(v_{avg} = \frac{v_0 + v}{2}\) To find the time (t) taken, use the equation: \(d = v_{avg} \times t\) Rearrange to find t: \(t = \frac{d}{v_{avg}}\) Plug in the values and determine t: \(t = \frac{15.0 \ \text{m}}{\frac{26.4 \ \text{m/s} + 23.76 \ \text{m/s}}{2}}\) \(t = 0.6151 \ \text{s}\)

Use the impulse-momentum theorem

The impulse-momentum theorem states that the impulse, which is the product of the net force acting on an object and the time interval, equals the change in momentum of the object: \(F_{\text {net }} \times \Delta t = \Delta p\) Since the only force acting against the motion of the softball is the force of air resistance, we can write the equation as: \(F_{\text {air }} \times \Delta t = m \times \Delta v\) Rearrange the equation to find the average force of air resistance, \(F_{\text {air }}\): \(F_{\text {air }} = \frac{m \times \Delta v}{\Delta t}\) Plug in the values and determine \(F_{\text {air }}\): \(F_{\text {air }} = \frac{0.250 \ \text{kg} \times (26.4 \ \text{m/s} - 23.76 \ \text{m/s})}{0.6151 \ \text{s}}\) \(F_{\text {air }} = 1.367 \ \text{N}\) The average force of air resistance exerted on the softball during its movement from the pitcher to the plate is approximately 1.367 N.

Key concepts

Impulse-Momentum Theorem

The impulse-momentum theorem is a fundamental principle in physics that describes the relationship between the force applied to an object, the time duration over which the force is applied, and the object’s change in momentum. Impulse is defined as the product of the average net force acting on an object and the time interval during which the force acts. The theorem can be formally expressed as:
\[\begin{equation}J = F_{\text{net}} \times \Delta t = \Delta p\end{equation}\]where:

• \(J\) is the impulse,
• \(F_{\text{net}}\) is the net force,
• \(\Delta t\) is the change in time,
• \(\Delta p\) is the change in momentum.

By applying this theorem to the problem, one calculates the average force when knowing the change in momentum and the time taken for this change. In the case of the softball, the change in velocity resulting from the air resistance essentially reflects a change in momentum over the given time.

Average Velocity Formula

The concept of average velocity is central to understanding motion in physics. It represents the average rate at which an object changes its position. The average velocity, denoted by \(v_{avg}\), over a time interval is calculated as the mean of the initial and final velocity. The formula is straightforward:
\[\begin{equation}v_{avg} = \frac{v_0 + v}{2}\end{equation}\]where:

• \(v_0\) is the initial velocity,
• \(v\) is the final velocity.

In our problem, finding the softball's average velocity is crucial to determine the time taken for the ball to reach the home plate, and subsequently using that time to calculate the force of air resistance.

Change in Momentum

Change in momentum, often symbolized by \(\Delta p\), is a key element in understanding the effects of forces on motion. Momentum itself is a measure of the 'quantity of motion' an object has and is calculated as the product of an object's mass and its velocity. The change in momentum is expressed as the difference between the final and initial momentum:\[\begin{equation}\Delta p = m \times (v - v_0)\end{equation}\]where:

• \(m\) is the mass of the object,
• \(v\) is the final velocity, and
• \(v_0\) is the initial velocity.

In our exercise, the softball experienced a reduction in speed due to air resistance, implying a negative change in momentum as the final velocity decreased from the initial velocity. This concept is vital for applying the impulse-momentum theorem to find the average force of air resistance.

Air Resistance Effects in Physics

Air resistance, also known as drag, is a force that opposes the motion of an object through the air. The effects of air resistance are particularly noticeable in objects moving at high speeds or with large surface areas. In physics, understanding air resistance is important for accurately predicting an object's motion, especially in cases where the resistance has a significant impact, such as our softball problem.As an object moves through air, it collides with air molecules, which exert a force opposite to the object's direction of motion. This force can depend on various factors, including the object's velocity, shape, size, and the properties of the air itself. When calculating the average force of air resistance on an object, one must consider the change in velocity of the object due to this force, as well as the time for which it acts, to apply the impulse-momentum theorem effectively.