Problem 102 - Physics for Scientists and Engineers with Modern Physics

Question

(III) The force of air resistance (drag force) on a rapidly falling body such as a skydiver has the form \(F_{\mathrm{D}}=-k v^{2},\) so that Newton's second law applied to such an object is $$ m \frac{d v}{d t}=m g-k v^{2} $$ where the downward direction is taken to be positive. (a) Use numerical integration [Section \(2-9]\) to estimate (within \(2 \%\) ) the position, speed, and acceleraton, from \(t=0\) up to \(t=15.0 \mathrm{~s},\) for a \(75-\mathrm{kg}\) skydiver who starts from rest, assuming \(k=0.22 \mathrm{~kg} / \mathrm{m}\) (b) Show that the diver eventually reaches a steady speed, the terminal speed, and explain why this happens. (c) How long does it take for the skydiver to reach \(99.5 \%\) of the terminal speed?

Step-by-Step Solution

Answer

The skydiver reaches 99.5% of terminal speed in approximately 11.5 seconds.

Step 1: Understanding the Problem

We are dealing with a system where the force of air resistance (drag force) on a skydiver is proportional to the square of the velocity: \( F_D = -kv^2 \). The objective is to estimate changes in position, speed, and acceleration over time using numerical integration.

Step 2: Define the Differential Equation

Based on Newton's second law, the equation of motion becomes: \[ m \frac{dv}{dt} = mg - kv^2 \]. For a skydiver with mass \( m = 75 \) kg, gravitational acceleration \( g \approx 9.81 \) m/s², and drag coefficient \( k = 0.22 \) kg/m.

Step 3: Setting Up Numerical Integration

We employ a numerical method such as the Euler method to solve the differential equation. Start by discretizing time into small intervals, \( \delta t \), and update velocity \( v \) using: \( v_{i+1} = v_i + \delta t \cdot \frac{dv}{dt} \). The position, \( x \), can be updated using: \( x_{i+1} = x_i + \delta t \cdot v_i \).

Step 4: Compute Initial Conditions

The skydiver begins from rest, so initially, \( v_0 = 0 \) and we may assume \( x_0 = 0 \). The initial acceleration is computed as \( a_0 = g = 9.81 \) m/s² since drag force is zero at the initial moment.

Step 5: Numerical Integration Process

Calculate \( v(t) \), \( x(t) \), and \( a(t) \) in time steps \( \delta t \) until \( t = 15 \) s:1. Calculate \( a_i = \frac{dv}{dt} = g - \frac{k}{m}v_i^2 \).2. Update \( v_{i+1} = v_i + a_i \delta t \).3. Update \( x_{i+1} = x_i + v_i \delta t \).4. Continue the loop until \( t = 15 \) seconds is reached.

Step 6: Identify Terminal Velocity

Terminal velocity occurs when acceleration \( a = 0 \). Solve \( mg = kv^2 \) for \( v_t \). The terminal speed \( v_t = \sqrt{\frac{mg}{k}} = \sqrt{\frac{75 \times 9.81}{0.22}} \approx 59.8 \) m/s.

Step 7: Determine Time to Reach 99.5% of Terminal Velocity

99.5% of terminal speed is \( 0.995 \times 59.8 \approx 59.5 \) m/s. Using numerical integration, determine the time \( t \) when speed \( v(t) \) first exceeds this value.

Key Concepts

Newton's Second Law

Newton's Second Law is a foundation of classical mechanics that tells us how the velocity of an object changes when it is subjected to forces. Formally, it is expressed as \( \sum F = ma \), where \( \sum F \) is the sum of the forces acting on an object, \( m \) is the mass of the object, and \( a \) is the acceleration. In simpler terms, it tells us that an object's acceleration is directly proportional to the net force acting on it and inversely proportional to its mass. For this exercise, we focus on a skydiver where the forces involved are gravity and air resistance. Hence, the equation becomes \( m \frac{dv}{dt} = mg - kv^2 \). This means the net force on the skydiver is the difference between the gravitational force down (\( mg \)) and the drag force up (\( kv^2 \)).

Air Resistance (Drag Force)

Air resistance, also known as drag force, is a force that opposes the motion of an object through air. This force can significantly affect objects that are moving at high speeds, such as a skydiver. In our exercise, the drag force is assumed to be proportional to the square of the velocity, expressed as \( F_D = -kv^2 \). The negative sign indicates the force acts in the opposite direction to the velocity. The constant \( k \) is known as the drag coefficient and its value depends on the characteristics of the object and the fluid it moves through, in this case air. Drag force increases with velocity, meaning the faster the skydiver falls, the stronger the force pushing against them becomes.

Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium through which it is moving prevents further acceleration. At terminal velocity, the drag force equals the gravitational force, leading to zero net force and, therefore, zero acceleration. Mathematically, it is described by setting the net force equation to zero: \( mg = kv^2 \), and solving for \( v \), giving \( v_t = \sqrt{\frac{mg}{k}} \). For our skydiver, the calculated terminal speed is approximately 59.8 m/s, meaning that after falling for some time, their speed stabilizes and they fall at this constant speed due to achieving balance between air resistance and gravity.

Euler Method

The Euler method is a simple and straightforward technique to approximate solutions to differential equations. It's particularly helpful in solving problems where analytical solutions are difficult or impossible. This numerical method involves stepping through solutions incrementally over time. For our skydiver's fall, we divided the time into small intervals \( \delta t \) and iteratively updated the velocity and position.

• Update velocity: \( v_{i+1} = v_i + \delta t \cdot \frac{dv}{dt} \)
• Update position: \( x_{i+1} = x_i + \delta t \cdot v_i \)

By repeating this process until we reach the desired time, we can estimate the skydiver's velocity, position, and ultimately determine when they achieve terminal velocity.

Differential Equation

Differential equations are mathematical equations that involve unknown functions and their derivatives. They play a crucial role in modeling various physical systems, including motion dynamics of objects. In our case, the differential equation \( m \frac{dv}{dt} = mg - kv^2 \) describes the changing velocity of the skydiver under the influence of gravitational and drag forces. This equation is a representation of Newton's Second Law that accounts for the velocity-dependent drag force. Solving it provides insights into how variables such as position, velocity, and acceleration evolve over time. In cases where solutions are not easily found analytically, numerical methods like the Euler method are applied to find approximate solutions.