The Physics
Hypertextbook
Opus in profectus

# Falling Bodies

## Discussion

### acceleration due to gravity

Want to see an object accelerate?

• Pick something up with your hand and drop it. When you release it from your hand, its speed is zero. On the way down its speed increases. The longer it falls the faster it travels. Sounds like acceleration to me.
• But acceleration is more than just increasing speed. Pick up this same object and toss it vertically into the air. On the way up its speed will decrease until it stops and reverses direction. Decreasing speed is also considered acceleration.
• But acceleration is more than just changing speed. Pick up your battered object and launch it one last time. This time throw it horizontally and notice how its horizontal velocity gradually becomes more and more vertical. Since acceleration is the rate of change of velocity with time and velocity is a vector quantity, this change in direction is also considered acceleration.

In each of these examples the acceleration was the result of gravity. Your object was accelerating because gravity was pulling it down. Even the object tossed straight up is falling — and it begins falling the minute it leaves your hand. If it wasn't, it would have continued moving away from you in a straight line. This is the acceleration due to gravity.

What are the factors that affect this acceleration due to gravity? If you were to ask this of a typical person, they would most likely say "weight" by which the actually mean "mass" (more on this later). That is, heavy objects fall fast and light objects fall slow. Although this may seem true on first inspection, it doesn't answer my original question. "What are the factors that affect the acceleration due to gravity?" Mass does not affect the acceleration due to gravity in any measurable way. The two quantities are independent of one another. Light objects accelerate more slowly than heavy objects only when forces other than gravity are also at work. When this happens, an object may be falling, but it is not in free fall. Free fall occurs whenever an object is acted upon by gravity alone.

Try this experiment.

• Obtain a piece of paper and a pencil. Hold them at the same height above a level surface and drop them simultaneously. The acceleration of the pencil is noticeably greater than the acceleration of the piece of paper, which flutters and drifts about on its way down.

Something else is getting in the way here — and that thing is air resistance (also known as aerodynamic drag). If we could somehow reduce this drag we'd have a real experiment. No problem.

• Repeat the experiment, but before you begin, wad the piece of paper up into the tightest ball possible. Now when the paper and pencil are released, it should be obvious that their accelerations are identical (or at least more similar than before).

We're getting closer to the essence of this problem. If only somehow we could eliminate air resistance altogether. The only way to do that is to drop the objects in a vacuum. It is possible to do this in the classroom with a vacuum pump and a sealed column of air. Under such conditions, a coin and a feather can be shown to accelerate at the same rate. (In the olden days in Great Britain, a guinea coin was used and so this demonstration is sometimes still called the "guinea and feather".) A more dramatic demonstration was done on the surface of the moon — which is as close to a true vacuum as humans are likely to experience any time soon. Astronaut David Scott released a rock hammer and a falcon feather at the same time during the Apollo 15 lunar mission in 1971. In accordance with the theory I am about to present, the two objects landed on the lunar surface simultaneously (or very nearly so). Only an object in free fall will experience a pure acceleration due to gravity.

### the leaning tower of Pisa

Let's jump back in time for a bit. In the Western world prior to the Sixteenth Century, it was generally assumed that the acceleration of a falling body would be proportional to its mass — that is, a 10 kg object was expected to accelerate ten times faster than a 1 kg object. The ancient Greek philosopher Aristotle of Stagira (384–322 BCE), included this rule in what was perhaps the first book on mechanics. It was an immensely popular work among academicians and over the centuries it had acquired a certain devotion verging on the religious. It wasn't until the Italian scientist Galileo Galilei (1564–1642) came along that anyone put Aristotle's theories to the test. Unlike everyone else up to that point, Galileo actually tried to verify his own theories through experimentation and careful observation. He then combined the results of these experiments with mathematical analysis in a method that was totally new at the time, but is now generally recognized as the way science gets done. For the invention of this method, Galileo is generally regarded as the world's first scientist.

In a tale that may be apocryphal, Galileo (or an assistant, more likely) dropped two objects of unequal mass from the Leaning Tower of Pisa. Quite contrary to the teachings of Aristotle, the two objects struck the ground simultaneously (or very nearly so). Given the speed at which such a fall would occur, it is doubtful that Galileo could have extracted much information from this experiment. Most of his observations of falling bodies were really of bodies rolling down ramps. This slowed things down enough to the point where he was able to measure the time intervals with water clocks and his own pulse (stopwatches and photogates having not yet been invented). This he repeated "a full hundred times" until he had achieved "an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse beat."

With results like that, you'd think the universities of Europe would have conferred upon Galileo their highest honor, but such was not the case. Professors at the time were appalled by Galileo's comparatively vulgar methods even going so far as to refuse to acknowledge that which anyone could see with their own eyes. In a move that any thinking person would now find ridiculous, Galileo's method of controlled observation was considered inferior to pure reason. Imagine that! I could say the sky was green and as long as I presented a better argument than anyone else, it would be accepted as fact contrary to the observation of nearly every sighted person on the planet.

Galileo called his method "new" and wrote a book called Discourses on Two New Sciences wherein he used the combination of experimental observation and mathematical reasoning to explain such things as one dimensional motion with constant acceleration, the acceleration due to gravity, the behavior of projectiles, the speed of light, the nature of infinity, the physics of music, and the strength of materials. His conclusions on the acceleration due to gravity were that…

the variation of speed in air between balls of gold, lead, copper, porphyry, and other heavy materials is so slight that in a fall of 100 cubits a ball of gold would surely not outstrip one of copper by as much as four fingers. Having observed this I came to the conclusion that in a medium totally devoid of resistance all bodies would fall with the same speed.

For I think no one believes that swimming or flying can be accomplished in a manner simpler or easier than that instinctively employed by fishes and birds. When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody?

I greatly doubt that Aristotle ever tested by experiment.

Galileo Galilei, 1638

Despite that last quote, Galileo was not immune to using reason as a means to validate his hypothesis. In essence, his argument ran as follows. Imagine two rocks, one large and one small. Since they are of unequal mass they will accelerate at different rates — the large rock will accelerate faster than the small rock. Now place the small rock on top of the large rock. What will happen? According to Aristotle, the large rock will rush away from the small rock. What if we reverse the order and place the small rock below the large rock? It seems we should reason that two objects together should have a lower acceleration. The small rock would get in the way and slow the large rock down. But two objects together are heavier than either by itself and so we should also reason that they will have a greater acceleration. This is a contradiction.

Here's another thought problem. Take two objects of equal mass. According to Aristotle, they should accelerate at the same rate. Now tie them together with a light piece of string. Together, they should have twice their original acceleration. But how do they know to do this? How do inanimate objects know that they are connected? Let's extend the problem. Isn't every heavy object merely an assembly of lighter parts stuck together? How can a collection of light parts, each moving with a small acceleration, suddenly accelerate rapidly once joined? We've argued Aristotle into a corner. The acceleration due to gravity is independent of mass.

Galileo made plenty of measurements related to the acceleration due to gravity but never once calculated its value (or if he did, I have never seen it reported anywhere). Instead he stated his findings as a set of proportions and geometric relationships — lots of them. His description of constant speed required one definition, four axioms, and six theorems. All of these relationships can now be written as the single equation in modern notation.

 v̅ = Δs Δt

Algebraic symbols can contain as much information as several sentences of text, which is why they are used. Contrary to the common wisdom, mathematics makes life easier.

### location, location, location

The generally accepted value is…

g = 9.8 m/s2

or in non-SI units…

g = 35 kph/s = 22 mph/s = 32 feet/s2

It is useful to memorize this number (as millions of people around the globe already have), however, it should also be pointed out that this number is not a constant. Although mass has no effect on the acceleration due to gravity, there are other factors that do.

Everyone reading this should be familiar with the images of the astronauts hopping about on the moon and should know that the gravity there is weaker than it is on the Earth — about one sixth as strong or approximately 1.6 m/s2. That is why the astronauts were able to hop around on the surface easily despite the weight of their space suits. In contrast, gravity on Jupiter is stronger than it is on the Earth — about two and a half times stronger or 25 m/s2. Astronauts cruising through the top of Jupiter's thick atmosphere would find themselves struggling to stand up inside their space ship. The acceleration due to gravity varies with location.

Furthermore, even on the Earth, this value varies with latitude and altitude (to be discussed in later chapters). The acceleration due to gravity is greater at the poles than at the equator and greater at sea level than atop Mount Everest. There are also local variations that depend upon geology. The value of 9.8 m/s2 is thus merely a convenient average over the entire surface of the Earth. This value is accurate to two significant digits up to the altitude at which commercial jets fly (18 km, 29,000 feet, or 5.5 miles). The acceleration due to gravity is effectively 9.8 m/s2 over the entire surface of the Earth.

How crazy are you for accuracy? For most applications, the value of 9.8 m/s2 is more than sufficient. If you're in a hurry, or don't have access to a calculator, or just don't need to be that accurate; rounding g to 10 m/s2 is often acceptable. During a multiple choice exam where calculators aren't allowed, this is often the way to go. If you need greater accuracy, consult a comprehensive reference work to find the accepted value for your latitude and altitude.

If that's not good enough, then obtain the required instruments and measure the local value to as many significant digits as you can. You may learn something interesting about your location. I once met a geologist whose job it was to measure g across a portion of West Africa. When I asked him who he worked for and why he was doing this, he basically refused to answer other than to say that one could infer the interior structure of the Earth from a gravimetric map prepared from his findings. Knowing this, one might then be able to identify structures where valuable minerals or petroleum might be found.

Like all professions, those in the gravity measuring business (gravimetry) have their own special jargon. The SI unit of acceleration is the meter per second squared [m/s2]. Split that into a hundred parts and you get the centimeter per second squared [cm/s2] also known as the gal [Gal] in honor of Galileo. (Note that the unit is written in lowercase as a word but is capitalized as a symbol.) Split that into a thousand parts and you get a milligal [mGal]. Since earth's gravity produces a surface acceleration of about 10 m/s2, a milligal is about 1 millionth of the value we're all used to. Measurements with this precision can be used to study changes in the Earth's crust, sea levels, ocean currents, polar ice, and groundwater. Push it a little bit further and it should be possible to detect changes in earth's atmosphere. Gravity is a heavy subject that will be discussed in more detail later in this book.

### Gee, Wally

As was discussed earlier, don't confuse the phenomena of acceleration due to gravity with the unit of the same name. While the quantity g has a value that depends on location and is approximately 9.8 m/s2 on earth, the unit gravity has the defined value of…

g = 9.80665 m/s2

You may also have noticed they use slightly different symbols. The unit uses the roman or upright g while the natural phenomena uses the italic or oblique g. Don't confuse g with g.

The unit g is often used to measure the acceleration of a reference frame. "Say what?" This is technical language that will be elaborated upon later in another section of this book, but I will explain it with examples for now. As I write this, I'm sitting in front of my computer in my home office. Gravity is drawing my body down into my office chair, my arms toward the desk, and my fingers toward the keyboard. This is the normal 1 g (one gee) world we're all accustomed to. I could take a laptop computer with me to an amusement park, get on a roller coaster, and try to get some writing done there. Gravity works on a roller coaster just as it does at home, but since the roller coaster is accelerating up and down (not to mention side to side) the sensation of normal earth gravity is lost. There will be times when I feel heavier than normal and times when I fell lighter than normal. These correspond to periods of more than one g and less than one g. I could also take my laptop with me on a trip to outer space. After a brief period of 2 or 3 g (two or three gees) accelerating away from the surface of the Earth, most space journeys are spent in conditions of apparent weightlessness or 0 g (zero gee). This happens not because gravity stops working (gravity has infinite range and is never repulsive), but because a spacecraft is an accelerating reference frame. As I said earlier, this concept will be discussed more thoroughly in a later section of this book.