Newton's Second Law and Air Resistance: Falling Objects

Newton's Second Law of Motion

Introduction to Newton's Second Law

Newton's Second Law of Motion describes the relationship between the net force acting on an object, its mass, and the resulting acceleration. This law is fundamental in analyzing the motion of objects under various forces, including gravity and air resistance.

• Definition: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

• Formula: $F_{net} = m a$

• Application: Used to calculate the acceleration of objects when multiple forces are present.

Forces Acting on a Falling Object

Gravity and Air Resistance

When an object falls through the atmosphere, it is subject to two main forces: the force of gravity pulling it downward and the force of air resistance acting upward.

• Force of Gravity ($F_{gravity}$): The weight of the object, calculated as $F_{gravity} = m g$.

• Air Resistance ($F_{air}$): A force that opposes the motion of the object through the air, increasing with speed.

• Net Force ($F_{net}$): The difference between the downward force of gravity and the upward force of air resistance: $F_{net} = F_{gravity} - F_{air}$.

Step-by-Step Calculation: Acceleration of a Falling Object with Air Resistance

Example Problem

Consider a 20 N falling object encountering 5 N of air resistance. Calculate its acceleration.

1. Determine the Net Force:

   • $F_{net} = F_{gravity} - F_{air}$

   • $F_{net} = 20\,\text{N} - 5\,\text{N} = 15\,\text{N}$

2. Determine the Mass of the Object:

   • Weight ($W$) is related to mass ($m$) by $W = m g$

   • Given $W = 20\,\text{N}$ and $g = 9.8\,\text{m/s}^2$

   • $m = \frac{W}{g} = \frac{20\,\text{N}}{9.8\,\text{m/s}^2} \approx 2.04\,\text{kg}$

3. Calculate the Acceleration:

   • Using Newton's Second Law: $F_{net} = m a \implies a = \frac{F_{net}}{m}$

   • $a = \frac{15\,\text{N}}{2.04\,\text{kg}} \approx 7.35\,\text{m/s}^2$

Conclusion: The acceleration of the object is less than the acceleration due to gravity because of the opposing force of air resistance.

Terminal Velocity

Concept of Terminal Velocity

As a falling object's speed increases, the air resistance increases until it equals the force of gravity. At this point, the net force becomes zero, and the object continues to fall at a constant speed known as terminal velocity.

• Terminal Velocity: The constant speed achieved when the force of air resistance equals the force of gravity.

• Equation: $F_{gravity} = F_{air}$, so $a = 0$

• Application: Skydivers reach terminal velocity after a period of free fall, after which they no longer accelerate.

Summary Table: Forces and Acceleration in Free Fall with Air Resistance

Quantity	Symbol	Formula	Example Value
Weight	W	$W = m g$	20 N
Mass	m	$m = \frac{W}{g}$	2.04 kg
Net Force	$F_{net}$	$F_{gravity} - F_{air}$	15 N
Acceleration	a	$a = \frac{F_{net}}{m}$	7.35 m/s2

Key Concepts and Definitions

• Free Fall: Motion of an object under the influence of gravity alone, without air resistance.

• Air Resistance: The force exerted by air against the motion of a falling object; increases with speed.

• Net Force: The sum of all forces acting on an object; determines acceleration.

• Terminal Velocity: The constant speed reached when air resistance balances gravity.

Example Application

When a skydiver jumps from an airplane, they initially accelerate at $g = 9.8\,\text{m/s}^2$. As their speed increases, air resistance grows until it equals their weight, and they reach terminal velocity, falling at a constant speed.

Additional info: The notes also reference quiz questions and online resources, reinforcing the importance of understanding how air resistance affects acceleration and terminal velocity in real-world scenarios.