Falling Bodies and Physics
Falling bodies in physics refer to the behavior of objects under the influence of gravity, a force that attracts any two masses toward each other. The phenomenon of free fall occurs when an object is solely influenced by this gravitational force, typically resulting in uniform acceleration toward the Earth at approximately 9.8 meters per second squared. This interaction is affected by the object's mass, air resistance, and the gravitational pull of other bodies, such as the moon.
As objects fall, they experience terminal velocity, which is the maximum speed they reach when the force of gravity is balanced by air resistance, preventing further acceleration. The trajectory, or path, of falling or thrown objects varies depending on the frame of reference used for observation. Historical theories on falling bodies have evolved significantly, from Aristotle's early notions of mass and fall rates to Galileo's demonstration that all objects fall at the same rate, regardless of mass.
Isaac Newton later formulated the laws of gravitation, which were refined by Einstein's theory of general relativity, explaining gravity as the warping of space-time by mass. Understanding these principles not only illuminates the dynamics of falling bodies but also lays the groundwork for advancements in both classical and modern physics, including the ongoing exploration of quantum gravity.
Falling Bodies and Physics
FIELDS OF STUDY: Classical Mechanics
ABSTRACT: How and why things fall—fall down, arc through the air, or fall into orbit around one another—are simultaneously two of the most familiar and most abstract topics in physics. Isaac Newton was the first to write useful formulas to explain gravity and its effects on objects more accurately. In the early twentieth century, Albert Einstein’s theory of general relativity replaced Newton’s theory of gravity. However, Newton’s equations are still used often due to their relative simplicity and accuracy regarding everyday scales of measurement.
PRINCIPAL TERMS
- centripetal force: for an object moving in a uniform circle, the force directed toward the center of the circle.
- free fall: falling only under the influence of gravity.
- gravitational force: any two objects in the universe attract one another through a force proportionate to their mass. This attractive force is gravity.
- instantaneous velocity: the velocity (speed and direction of travel) of an object in motion at any one instant of time.
- reference frame: the velocity of an object relative to the objects around it and the point of observation.
- terminal velocity: the maximum velocity of an object in free fall, determined by the drag and buoyancy of the object relative to the force of gravity.
- trajectory: the path of a thrown or falling object, such as a baseball.
Gravity’s Influence: Falling Objects
Every object in the universe, from a Ping-Pong ball to a planet, exerts a gravitational force on objects around it. This gravitational force is proportional to the size (mass) of the object and the distance between it and the object it is acting on. Standing on Earth’s surface, an individual experiences a stronger gravitational force from Earth than Jupiter, which is nearly 590 million kilometers (370 million miles) away. Gravity is the source of an object’s weight. A scale is simply measuring the force with which an object is being pulled toward the gravitational source. In essence, objects (such as houses, people, even planes) are always falling toward the center of the earth, but the resistance of the crust stops them. This is why astronauts weigh so much less on the surface of the moon. The moon has much less mass than Earth and exerts less gravitational pull on the astronauts there. They are also so far from Earth that its gravity has a small effect on their weight as compared to the moon’s.
When an object falls under the force of Earth’s gravity, it accelerates at a uniform rate. This rate varies according to distance from Earth and resistance from air or another medium an object is moving through. It also changes under the gravitational pull of other bodies, such as the moon. However, for most purposes, a standard gravitational acceleration (written as g) of 9.8 meters per second per second (m/s2)—which represents an object in free fall at sea level—is sufficient.
A common free-fall myth insists that a penny dropped from the top of the Empire State Building will become a lethal projectile. This is because it is accelerating at 9.8 m/s2 for a distance of several hundred feet. Terminal velocity, however, limits the penny to a nonlethal (if painful) velocity. The faster an object falls, the more powerful the effect of drag from the air around it becomes. Eventually, the object reaches a point where the acceleration due to gravity is balanced by the deceleration due to drag. The acceleration becomes zero, and no matter how much longer the object falls, it will not speed up any further. The object’s instantaneous velocity (vf) is the speed and direction of its motion at a given moment. This is calculated as the product of the time spent (t) and the acceleration due to gravity (g).
An object dropped straight down is not, in absolute terms, moving in a straight line. The earth is rotating about its axis, and the entire planet is orbiting the sun. Therefore, the reference frame chosen for calculation is very important. English physicist Isaac Newton (1642–1727) assumed the stars in the sky were fixed points and made his calculations of motion relative to them. This method was good enough to lead to generally accurate predictions in most cases. However, scientists have since learned that the stars are not stationary at all. Similarly, for most calculations, it is safe to ignore Earth’s rotation and movement through space. If someone wants to determine the velocity of a penny dropped from the Empire State Building, he or she typically is not interested in its velocity relative to the rest of the universe—just relative to the building, the observer, and the street below.
The trajectory of a falling or thrown object is different depending on the frame of reference used to define it. If the frame of reference were set by two children playing catch on Earth’s surface, a thrown ball’s trajectory would be a simple arc. If the sun were set as the key reference point, then its trajectory would be complicated by the movement of the thrown ball around Earth and around the sun.
The History of Gravitational Theory
Theories to explain why and how objects fall are as old as antiquity. The ancient Greek philosopher Aristotle (ca. 384–ca. 322 BCE) theorized that more massive objects were pulled toward the center of the universe as a result of their innate heaviness. He predicted that heavy objects should fall faster than lighter ones.
In the seventeenth century CE, Galileo Galilei (1564–1642) found that objects fall at the same rate regardless of their mass, contrary to Aristotle’s ideas. Galileo is famously said to have dropped objects of various masses off the Leaning Tower of Pisa and recorded the time it took them to hit the ground, demonstrating that objects fell at the same rate regardless of their mass. Historians have since decided this was more likely a theoretical experiment rather than a physical one.
Later that same century, Newton codified the inverse-square law of gravitation. This law stated that the gravitational force exerted by one object on another is directly proportional to the object’s mass and inversely proportional to the square of the distance between the centers of the two objects. Newton’s equations were remarkably effective under most circumstances. However, when dealing with extremes (such as the orbit of Mercury), small errors in their predictions were found.
In 1915 Albert Einstein (1879–1955) introduced the theory of general relativity, which resolved these errors in prediction by proposing that gravity is the effect of mass and energy warping space-time. A massive object like the sun creates a large depression in space-time, causing smaller objects like the earth to move around it. So far, Einstein’s predictions have been very accurate. Newton’s theory has been replaced, but Newtonian gravity is still in common use for everyday calculations. Einstein’s theory is used mainly in cases of extreme gravity or when great accuracy is needed. When dealing with falling or thrown objects, it is generally safe to assume Newton’s laws apply.
Circular Motion and Centripetal Force
For objects moving in a circle—such as a ball being spun on a string or the moon orbiting the earth—centripetal force describes the force acting against the object’s momentum and pulling it toward the center of the circular path. In the case of orbits, gravity is a centripetal force.
This concept underpins the most practical idea for how to create artificial gravity in space. Consider a Tilt-a-Whirl at a fair: as the ride spins, the inside passenger feels heavier and pinned against the inside of the cart. A spacecraft spinning at a sufficient speed would generate a centripetal force that would mimic a planet’s gravity, even in the void of space. If the floor of the spacecraft were oriented toward the center of the spinning ring, it would be the astronaut’s feet that would be pinned, creating a feeling similar to gravity. However, the spacecraft would need to be larger than anything conceived of so far if it were to spin at a sustainable speed. It would also need to prevent the astronauts inside from experiencing a dizzying, potentially dangerous difference in the force acting on their feet versus their heads.
Sample Problem
Consider a baseball weighing 0.15 kilograms dropped from the top of a building. The baseball strikes the ground with a velocity of 20 meters per second. With this information, calculate the height of the building it was dropped from and the time it took to land.
Answer:
First, make note of what is known. The problem made no mention of extreme wind or other complicating factors. From this, it is safe to assume that simple Newtonian equations for acceleration due to gravity will suffice. Neither the object’s mass nor wind resistance will make a significant difference in the rate of its descent. Because the ball is dropped from a stationary point, it is safe to assume the trajectory it travels is a straight line down to the ground. Since everyday conditions are a valid assumption, the value of acceleration due to gravity (g)can be set to 9.8 m/s2.
Start by calculating (t). Using the formula for the instantaneous velocity (vf), plug in the velocity for the ball at impact and the acceleration due to gravity:
With a value for time (t), distance (d) fallen can be calculated using the following equation:
The Future of Gravitational Physics
Although its predictions have been proven by many experiments, the gravitational theory provided by general relativity is not perfect. In particular, it is incompatible with modern quantum mechanics. Attempts to translate general relativity into a quantum framework have proven incomplete at best. Thus, quantum gravity—the field of physics that attempts to describe gravity using the principles of quantum mechanics—is an area of much contention in modern theoretical physics. If the trajectory of gravitational theory across history is any indication, scientists will continue to refine gravitational theory so that it makes accurate predictions under increasingly extreme conditions.
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