FREE FALL ACCELERATION DUE TO GRAVITY—Aristotle to Galileo to Newton (F = M x A)

Posted on January 18th, 2019 by kramer

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 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 nearly so). Only an object in free fall will experience a pure acceleration due to gravity. I also showed the children a video of a NASA experiment in the country's largest vacuum chamber where they dropped a bowlingball and feathers in a vacuum and they accelerated at the exact same speed. The video discusses Galileo and Newton but concluded with one of Einstein's greatest discoveries.

https://www.youtube.com/watch?v=E43-CfukEgs

The Leaning Tower of Pisa

Let's jump back in time for a bit. In the Western world prior to the 16th 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 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.

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.

The Ratio of Force and Mass

Galileo got us this far, but we needed Isaac Newton (1642-1726) to take it one step further. Galileo was very interested in how things worked, while Newton was more interested in the why. His second law of motion states that acceleration is directly proportional to the net force and inversely proportional to the mass of the object.

Since acceleration is proportional to force, an increase in one will result in an increase in the other. More force, more acceleration. Less force, less acceleration. But because acceleration and mass are inversely proportional, this means that an increase in one will decrease the other. More mass means less acceleration, and less mass means more acceleration. So if we write this law as an equation, we get a = F/m, where a is the acceleration (usually in meters/second^2), F is net force in Newton, and m is mass in kilogram. Or we can write it as F=ma

Force = Mass times Acceleration Due to Gravity (9.8 m/s² or 32 ft./s²)

9.8 m/s²is equivalent to about 22 miles per hour (MPH) and our terminal velocity is about 120 MPH if falling with arms and legs stretched out like a skydiver (Felix Baumgartner holds the worlds record for free fall of 834 MPH; don’t try this at home kids). The video link to Felix's record breaking fall is at

https://www.youtube.com/watch?v=wPQpdYcZ23k

The longer version is at

https://www.youtube.com/watch?v=vvbN-cWe0A0

F = M x A 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.

This equation tells us that if the net force acting on an object is doubled, the acceleration of the object will also double. But if the mass is doubled, the acceleration will be halved. Finally, if both the net force and the mass are doubled, there will be no change in acceleration because the ratio of force to mass stays the same. 1/1 is the same as 2/2 - they both equal 1!

What does this have to do with free-fall? Well, it explains why in the absence of air resistance, heavier objects fall with the same acceleration as light ones. In fact, we even have a value for this acceleration: g, or 9.8 m/s^2. This is often rounded up to 10 m/s^2, and we'll use that for our calculations in this lesson for simplicity.

Where does this value come from? Say, for example, that we have a 1 kg person and a 1000 kg elephant. Ignoring that this is a very small person, it's a big difference in mass between the two, right? 1 kg is about 10 N, so when the 1000 kg elephant falls, the force due to gravity (its weight) is 10,000 N. For the 1 kg person, its weight is 10 N. The force is far greater for the elephant than the person, but its mass is also much greater. If we return to Newton's second law, we find that the acceleration for the elephant is 10,000 N / 1000 kg, which equals 10 m/s^2 (the unit of Newton can also be written as kg*m/s^2 so the kg of the force and mass cancel out).

What about the 1 kg person? If we do the math, we find that 10 N / 1 kg = 10 m/s^2 as well!

Can you see how the proportional force increases the acceleration while at the same time the inversely proportional mass decreases it? It's because of this relationship between force and mass that both objects will have the same acceleration in free-fall.

Your job this week: F (newtons) = M (weight kg) x A (acceleration g)

Read these pages on Free Fall, Acceleration Due to Gravity and Mass.
Take several objects of different weights that will not break when you drop them. Drop them two at a time (one in each hand) at the same time and watch (or have someone take a video; even in slow motion) the objects hit the ground at the same time.
Notice that when one of the objects creates a lot of drag, it does fall slower because there are forces at work other than gravity.
Calculate your Force in Newtons: (a) weigh yourself in pounds; (b) with a calculator or not, divide your pound weight by 2.2 or multiply it by .45 to get your weight in kilograms; (c) then multiply your weight in kilograms by Acceleration Due to Gravity on Earth of 9.8 m/s (or just 10 since it is so close; remember, just tack on a zero to multiply by 10) and that is your Force in Newtons.

Yourself Mother Father Sibling Friend

Weight in pounds _________ _________ _________ _________ ______

Weight in kilograms

Divide pounds by 2.2 _________ _________ _________ _________ ______

Force in Newtons

Multiply Weight in kilo-

grams by 9.8 m/s or 10 _________ _________ _________ _________ ______

Yourself Object B Object C Object D Obj. E