Question

A skydiver of mass \(82.3 \mathrm{~kg}\) (including outfit and equipment) floats downward suspended from her parachute, having reached terminal speed. The drag coefficient is 0.533 , and the area of her parachute is \(20.11 \mathrm{~m}^{2} .\) The density of air is \(1.14 \mathrm{~kg} / \mathrm{m}^{3}\). What is the air's drag force on her?

Short answer

Answer: The air's drag force on the skydiver when she reaches terminal speed is approximately 807.69 Newtons.

Step-by-step solution

Understanding the Drag force formula

The drag force formula is given by the equation: F_d = 0.5 * C_d * ρ * A * v^2 where F_d is the drag force, C_d is the drag coefficient, ρ( rho) is the air density, A is the object's reference area (parachute area in this case), and v is the velocity of the object. Since the skydiver has reached terminal speed, the drag force equals the gravitational force acting on her. Therefore, we can use Newton's second law of motion to write this relationship: F_d = m * g where m is the mass of the skydiver, and g is the gravitational acceleration (approximately 9.81 m/s^2). We will use these equations to find the drag force.

Calculating the drag force

We can now set the terminal speed drag force equal to the gravitational force: 0.5 * C_d * ρ * A * v^2 = m * g We have all the values except the velocity (v), so we'll first find the terminal velocity. Rearranging the equation for v: v^2 = (2 * m * g) / (C_d * ρ * A) Now, we can plug in the given values: v^2 = (2 * 82.3 * 9.81) / (0.533 * 1.14 * 20.11) Calculating the velocity: v^2 ≈ 127.14 v ≈ sqrt(127.14) ≈ 11.28 m/s Now that we have the terminal velocity, we can calculate the drag force using the drag force formula: F_d = 0.5 * C_d * ρ * A * v^2 F_d = 0.5 * 0.533 * 1.14 * 20.11 * (11.28)^2 Calculating the drag force: F_d ≈ 807.69 N So, the air's drag force on the skydiver is approximately 807.69 Newtons.

Key concepts

Understanding Drag Force

Drag force is a crucial concept when studying objects moving through a fluid like air. It represents the resistance or friction the object experiences. The magnitude of this force can be calculated using the formula:

\[ F_d = 0.5 \times C_d \times \rho \times A \times v^2 \]
Here:

• \( F_d \) is the drag force
• \( C_d \) is the drag coefficient, which quantifies the object's drag per unit area and depends on its shape
• \( \rho \) is the air density
• \( A \) is the reference area of the object
• \( v^2 \) is the square of the object's velocity

The drag force is proportional to the square of the velocity, meaning that as speed increases, drag force increases significantly. This formula helps us understand how various factors contribute to the overall resistance an object faces as it moves.

Exploring Terminal Velocity

Terminal velocity is the constant speed a falling object eventually reaches when the drag force equals the gravitational force acting on it. At terminal velocity, the object no longer accelerates and moves at a steady pace.

To find terminal velocity, we set the drag force equal to the gravitational force:

• Drag force: \( F_d = 0.5 \times C_d \times \rho \times A \times v^2 \)
• Gravitational force: \( m \times g \)

Hence, when these forces balance out, the formula simplifies to:

\[ v^2 = \frac{2 \times m \times g}{C_d \times \rho \times A} \]
This shows that terminal velocity is affected by:

• Mass of the object
• Gravitational acceleration (usually \(9.81 \text{ m/s}^2\) on Earth)
• Drag coefficient and air density
• Reference area of the object

Calculating terminal velocity gives insight into how these factors interplay and affect the descent speed.

Applying Newton's Second Law

Newton's Second Law of Motion is central in understanding dynamics of falling objects. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration:

\[ F = m \times a \]
In our scenario with the skydiver, when reaching terminal velocity, the net force becomes zero. This happens because the downward gravitational force is balanced by the upward drag force:

• Gravitational Force: \( F_g = m \times g \)
• Drag Force: \( F_d = 0.5 \times C_d \times \rho \times A \times v^2 \)

Therefore, \( F_g = F_d \), meaning the diver floats downward at a steady speed, with no acceleration. Newton's Second Law allows us to equate and solve for unknown variables based on the balance of forces, key in calculating terminal velocity and ensuring safe skydiving dynamics.

The Dynamics of Parachuting

Parachute dynamics is not just about a gentle landing; it's a fascinating interplay of physics that ensures safety from high altitudes. A parachute, when deployed, greatly increases the surface area exposed to airflow:

• This increased area significantly boosts the drag force.
• More drag leads to a dramatic reduction in descent speed, helping the skydiver achieve terminal velocity safely.

The parachute's drag coefficient, usually larger than that of a free-falling object, is vital in the calculation of descending speed. The combination of a large reference area and suitable drag coefficient ensures that the drag force matches gravitational pull sooner. Thus, skydivers can reach safe terminal velocities quickly.

Understanding parachute dynamics involves evaluating the balance between upward drag and downward gravitational forces. A well-designed parachute ensures the diver lands safely by managing these forces efficiently.