Newton's Law of Free Fall - Free Source Library

The Law of Free Fall According to Newton

Isaac Newton, one of the most prominent figures in the history of physics, made groundbreaking contributions to our understanding of the natural world. Among his many discoveries, the law of free fall stands out as a key principle in classical mechanics. The concept of free fall is essential to understanding how objects move when influenced solely by gravity, without any other forces, such as air resistance, acting upon them. Newton’s formulation of the law of free fall was crucial in advancing the understanding of gravitational forces and the motion of bodies on Earth and in space.

What is Free Fall?

Free fall refers to the motion of an object under the influence of gravity alone, without any resistance from other forces like air resistance. In a free-falling state, the only force acting on an object is gravity. This occurs when an object is dropped from a certain height or thrown vertically upward or downward in the absence of any other significant forces.

When Newton developed his theory of free fall, he built on the work of earlier scientists, such as Galileo Galilei, who had demonstrated that objects fall at the same rate in a vacuum, regardless of their mass. However, it was Newton who provided a more comprehensive and mathematical description of how gravity influences the motion of objects.

Newton’s Law of Universal Gravitation

To understand Newton’s law of free fall, it is essential first to recognize his more general theory of gravitation, known as the law of universal gravitation. According to Newton, every mass in the universe exerts an attractive force on every other mass. This force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Mathematically, the law of universal gravitation is expressed as:

$F = G \frac{m_1 m_2}{r^2}$

Where:

$F$ is the gravitational force between two objects.
$G$ is the gravitational constant ( $6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$ ).
$m_1$ and $m_2$ are the masses of the two objects.
$r$ is the distance between the centers of the two objects.

This universal law explains the force that pulls objects toward each other, and it is this force that governs the motion of freely falling objects.

Free Fall and Gravitational Acceleration

In the context of free fall near Earth’s surface, all objects experience the same acceleration due to gravity, regardless of their mass. This is one of the key insights that Newton’s law of free fall helped to confirm. Galileo’s earlier experiments suggested that in the absence of air resistance, all objects fall at the same rate. Newton expanded on this idea by formalizing it in terms of gravitational acceleration.

The acceleration due to gravity on Earth is denoted as $g$ , and its average value near Earth’s surface is approximately:

$g \approx 9.81 \, \text{m/s}^2$

This means that, in free fall, the velocity of an object increases by approximately 9.81 meters per second every second. The acceleration is constant for all objects, regardless of their mass, in the absence of other forces such as air resistance.

The Equations of Motion for Free-Falling Objects

To describe the motion of objects in free fall, we use the basic equations of motion from Newtonian mechanics. These equations relate displacement, velocity, acceleration, and time. For an object in free fall, the acceleration is constant and equal to $g$ , so we can apply the following kinematic equations:

Velocity as a function of time:

$v = u + g t$

Where:

$v$ is the final velocity of the object.
$u$ is the initial velocity (which is zero if the object is simply dropped).
$g$ is the acceleration due to gravity.
$t$ is the time the object has been falling.

Displacement as a function of time:

$s = u t + \frac{1}{2} g t^2$

Where:

$s$ is the displacement (the distance fallen).
$u$ is the initial velocity (which is zero if the object starts from rest).
$g$ is the acceleration due to gravity.
$t$ is the time elapsed.

If the object starts from rest (i.e., $u = 0$ ), these equations simplify to:

$v = g t$
$s = \frac{1}{2} g t^2$

These equations describe the fundamental principles of motion for objects in free fall. For instance, if an object is dropped from a height, we can use these equations to calculate how fast it will be moving when it hits the ground or how far it will fall in a given amount of time.

The Influence of Air Resistance

In reality, most objects do not fall in a perfect vacuum, and air resistance plays a role in their motion. When objects fall through the air, they experience a drag force that opposes their motion, slowing their fall. The magnitude of this resistance depends on factors such as the object’s speed, size, shape, and the density of the air.

Air resistance causes objects to reach a terminal velocity, which is the constant speed at which the force of gravity is balanced by the drag force. At this point, the object stops accelerating and falls at a constant velocity. For example, a skydiver falling through the atmosphere will eventually reach terminal velocity, which can be around 53 m/s (about 120 mph) in a belly-to-earth position.

However, in the context of Newton’s law of free fall, we assume that the only force acting on the object is gravity, meaning that air resistance is neglected. This is an idealized scenario, often used for simplicity in basic physics problems.

Free Fall on Other Celestial Bodies

While the concept of free fall is most commonly associated with objects falling on Earth, it applies universally to any object near any celestial body with mass, such as the Moon, Mars, or even Jupiter. However, the value of gravitational acceleration ( $g$ ) varies depending on the mass and radius of the celestial body in question.

For example, on the Moon, the acceleration due to gravity is only about 1/6th of that on Earth, approximately $1.625 \, \text{m/s}^2$ . This means that objects on the Moon fall more slowly than those on Earth. Similarly, the acceleration due to gravity on Mars is about $3.7 \, \text{m/s}^2$ , which is weaker than Earth’s gravity but stronger than the Moon’s.

The principle of free fall is universal and applies to all objects, but the actual rate of fall is determined by the gravitational field of the specific celestial body.

Conclusion

Newton’s law of free fall, based on his broader theory of universal gravitation, remains one of the cornerstones of classical mechanics. It provides a comprehensive framework for understanding how objects move under the influence of gravity alone, and it is instrumental in describing the motion of objects both on Earth and in space. Through the application of Newton’s laws, we can predict how fast an object will fall, how far it will travel, and how long it will take to reach the ground, all while considering the effect of gravity and its interaction with mass.

As modern physics has progressed, particularly with the advent of general relativity and quantum mechanics, the framework for understanding gravity has expanded. Yet, Newton’s description of free fall and gravity remains an essential and accurate approximation for many everyday situations, such as predicting the motion of falling objects, the trajectory of projectiles, or the motion of planets within our solar system. His work laid the foundation for centuries of scientific discovery and continues to inspire further exploration into the mysteries of the universe.