Question

The drag force, due to air resistance, on an object can be expressed as $$ F_{d}=\frac{1}{2} C_{D} \varrho A V^{2} $$ where \(\varrho\) is the density of the air, \(V\) is the velocity of the object, \(A\) is the cross-sectional area (normal to the velocity direction), and \(C_{D}\) is the drag coefficient, which depends heavily on the shape of the object and the roughness of the surface. The gravity force on an object with mass \(m\) is \(F_{g}=m g\), where \(g=9.81 \mathrm{~ms}^{-2}\). We can use the formulas for \(F_{d}\) and \(F_{g}\) to study the importance of air resistance versus gravity when kicking a football. The density of air is \(\varrho=1.2 \mathrm{~kg} \mathrm{~m}^{-3}\). We have \(A=\pi a^{2}\) for any ball with radius \(a\). For a football \(a=11 \mathrm{~cm}\). The mass of a football is \(0.43 \mathrm{~kg}, C_{D}\) can be taken as \(0.2\). Make a program that computes the drag force and the gravity force on a football. Write out the forces with one decimal in units of Newton \(\left(\mathrm{N}=\mathrm{kg} \mathrm{m} / \mathrm{s}^{2}\right)\). Also print the ratio of the drag force and the gravity force. Define \(C_{D}, \varrho, A, V, m, g, F_{d}\), and \(F_{g}\) as variables, and put a comment with the corresponding unit. Use the program to calculate the forces on the ball for a hard kick, \(V=120 \mathrm{~km} / \mathrm{h}\) and for a soft kick, \(V=10 \mathrm{~km} / \mathrm{h}\) (it is easy to mix inconsistent units, so make sure you compute with \(V\) expressed in \(\mathrm{m} / \mathrm{s}\) ). Name of program file: kick.py.

Short answer

Drag force (hard): 4.6 N, Gravity force: 4.2 N, Ratio: 1.08. Drag force (soft): 0.03 N, Ratio: 0.01.

Step-by-step solution

Convert Velocity Units

Given velocities are in km/h. First, we need to convert these velocities to m/s, as SI units demand consistent units for calculation. For the hard kick:\[ V_{hard} = \frac{120\,\text{km/h} \times 1000\,\text{m/km}}{3600\,\text{s/h}} = 33.33\,\text{m/s} \]For the soft kick:\[ V_{soft} = \frac{10\,\text{km/h} \times 1000\,\text{m/km}}{3600\,\text{s/h}} = 2.78\,\text{m/s} \]

Define Constants and Variables

Define all constants and variables required for the calculation:- \( C_D = 0.2 \) (dimensionless)- \( \varrho = 1.2\,\text{kg/m}^3 \) (density of air)- \( a = 0.11\,\text{m} \) (radius of the football, converted to meters)- \( m = 0.43\,\text{kg} \) (mass of the football)- \( g = 9.81\,\text{m/s}^2 \) (acceleration due to gravity)- \( A = \pi a^2 = \pi \times (0.11)^2 \) (cross-sectional area of the football)

Calculate Drag Force

Using the formula \( F_d = \frac{1}{2} C_D \varrho A V^2 \), calculate the drag force for both kicks:Hard kick: \[ F_{d,hard} = \frac{1}{2} \times 0.2 \times 1.2 \times \pi \times (0.11)^2 \times (33.33)^2 = 4.57\,\text{N} \]Soft kick: \[ F_{d,soft} = \frac{1}{2} \times 0.2 \times 1.2 \times \pi \times (0.11)^2 \times (2.78)^2 = 0.03\,\text{N} \]

Calculate Gravity Force

Gravity force is constant for both kicks:\[ F_g = m \times g = 0.43 \times 9.81 = 4.22\,\text{N} \]

Calculate Force Ratios

Calculate the ratio of drag force to gravity force for both kicks:Hard kick:\[ \frac{F_{d,hard}}{F_g} = \frac{4.57}{4.22} = 1.08 \]Soft kick:\[ \frac{F_{d,soft}}{F_g} = \frac{0.03}{4.22} = 0.01 \]

Write the Program

Create a Python script named `kick.py` that encapsulates the above calculations. Ensure the program outputs the drag force, gravity force, and their ratio for both kicking scenarios. ```python # Constants C_D = 0.2 # Drag coefficient (unitless) varrho = 1.2 # Air density (kg/m^3) pi = 3.14159 radius = 0.11 # Radius of the football (m) mass = 0.43 # Mass of the football (kg) g = 9.81 # Gravitational acceleration (m/s^2) # Derived quantities A = pi * radius**2 # Cross-sectional area (m^2) # Velocities in m/s V_hard = 120 * 1000 / 3600 # Hard kick (m/s) V_soft = 10 * 1000 / 3600 # Soft kick (m/s) # Drag forces F_d_hard = 0.5 * C_D * varrho * A * V_hard**2 F_d_soft = 0.5 * C_D * varrho * A * V_soft**2 # Gravity force F_g = mass * g # Output results print(f"Drag force for hard kick: {F_d_hard:.1f} N") print(f"Drag force for soft kick: {F_d_soft:.1f} N") print(f"Gravity force: {F_g:.1f} N") print(f"Ratio of drag to gravity force (hard kick): {F_d_hard/F_g:.2f}") print(f"Ratio of drag to gravity force (soft kick): {F_d_soft/F_g:.2f}") ```

Key concepts

Drag Force Calculation

Drag force is a kind of resistance force that opposes the motion of an object through a fluid, like air. It's important when considering an object's dynamics, such as a football being kicked. The drag force experienced by an object can be expressed mathematically as:\[ F_{d} = \frac{1}{2} C_{D} \varrho A V^{2} \]Where:

• \( C_{D} \) is the drag coefficient, a measure of an object's resistance to fluid flow. It's dimensionless and depends on the object's shape and surface roughness.
• \( \varrho \) represents the air density, which is approximately \( 1.2 \text{ kg/m}^3 \) at sea level under normal conditions.
• \( A \) is the cross-sectional area of the object, calculated using its radius for a spherical shape, e.g., a football . \( A = \pi a^{2} \) where \( a \) is the radius. For a football, this would be \( 0.11 \text{ m} \) (converted from centimeters).
• \( V \) is the velocity of the object. It's important to ensure that this value is in meters per second (m/s) for consistent unit calculations.

Using this formula, you can calculate the drag force exerted on the football at different speeds, such as during a hard or soft kick.

Gravity Force

The force of gravity acts on all objects with mass, pulling them towards the Earth. For a football or any other object, the gravity force can be calculated using the formula:\[ F_{g} = m g \]Here:

• \( m \) refers to the mass of the object—in our example, the mass of the football is \( 0.43 \text{ kg} \).
• \( g \) is the acceleration due to gravity, a constant value of approximately \( 9.81 \text{ m/s}^2 \) on Earth's surface.

The gravity force is constant regardless of the kicking velocity because it depends only on mass and gravitational acceleration, not on the speed or direction of the object. This makes it a crucial factor when comparing it with varying forces like drag during different kicking scenarios.

Velocity Conversion

Converting velocity measurements is an important step to ensure accuracy in calculations involving different unit systems. In the exercise, velocities were presented in kilometers per hour (km/h), but we need them in meters per second (m/s) for physics calculations.To convert velocity from km/h to m/s, use:\[ V_{\text{m/s}} = \frac{V_{\text{km/h}} \times 1000}{3600} \]This formula divides the velocity in km/h by a factor of \(3.6\), effectively converting kilometers to meters and hours to seconds.

• For a hard kick at \(120 \text{ km/h}\), the velocity becomes \(33.33 \text{ m/s} \).
• For a soft kick at \(10 \text{ km/h}\), it converts to \(2.78 \text{ m/s} \).

By standardizing the units to meters per second, subsequent calculations for drag and gravity forces can be performed accurately and consistently.

Python Script Writing

Writing a Python script to compute drag and gravity forces involves defining variables for fixed and calculated values, performing arithmetic operations, and outputting the results. This approach automates calculations and ensures precision.Start with defining constants and variables:

• The drag coefficient \( C_D = 0.2 \),
• The density of air \( \varrho = 1.2 \text{ kg/m}^3 \),
• Radius \( a = 0.11 \text{ m}\),
• Mass \( m = 0.43 \text{ kg}\).

Use these to calculate derived quantities, like the cross-sectional area \( A = \pi a^2 \).Next, determine the drag forces using the formula for both velocities, and compute the gravitational force for comparison.The Python script outputs results clearly, specifying:

• The drag force for both kicks,
• The gravitational force,
• The ratio of the drag force to gravity force for easy interpretation.

This program not only simplifies calculations but also helps in understanding the relationship between these forces by demonstrating their ratios in a practical scenario.