Many scholars of ancient times, including Aristotle (ca. 350 BCE), believed that heavier objects fall through the air with faster acceleration than lighter objects. It is hard to detect whether or not this is true with real-life experimentation.

Try dropping a lacrosse ball and a tennis ball (they are close to the same size) simultaneously from the same height. Can you tell if one hits the ground before the other?

Nonetheless, it does seem reasonable to suspect that gravity has a “greater effect” on heavier objects causing them to speed up faster than lighter objects when they fall. Many scholars reasoned this.

But Galileo from the 1500s questioned this reasoning.

He said to imagine two objects of the same size and shape, but of different masses, being dropped from the Leaning Tower of Pisa at the same time. If gravity has stronger effect on the heavier object, then it would land first, taking the shorter amount of time to reach the ground.

Now attach a very light string between the two objects, ostensibly making them one object which is of greater mass than either individual object. Gravity then should have an even stronger effect on this system. So the two linked objects dropped from the Tower should fall to the ground in shorter time still. But how do the two objects now “know” they are attached as one object and should fall faster? This doesn’t make sense!

Galileo concluded then that all objects of the same size and shape must fall through the air in unison, at the same shared rate of acceleration, irrespective of their masses. (Air resistance causes objects to fall at different rates according to their shapes: a sheet of paper is slower to fall than a paper-clip of the same mass, for instance. But it is not gravity causing this variation.)

Letters and biographies about Galileo written at and near the time say that he considered going to the top of the Leaning Tower of Pisa to drop and time falling objects so as to verify that their accelerations to the ground were in unison. But there is no evidence to suggest that he actually did this. (But he did mimic the experiment by rolling objects down ramps. Their slower motion made is possible to time their rates of descent due to gravity.)

Aside: In 1971, Apollo 15 Commander David Scott while walking the surface of the Moon performed a live television demonstration in which he simultaneous dropped a hammer and a feather to see if, in the absence of an atmosphere, they would indeed accelerate to the ground at identical rates. You can see the video here.

What Galileo Saw

Let’s imagine Galileo did drop objects from the top of the tower and could time their falls.

Here’s a completely made-up data set that assumes the tower is 100 feet tall and that one object took 2.5 seconds to fall to the ground.  (Is this even close to being realistic?)

Here \(t\) represents the time passed (measured in seconds) since dropping the object and \(h\) the height of the object at each time (measured in feet).

What should data like this reveal about acceleration due to gravity?

Well,

velocity is the rate of change of position (in our case, heights)

acceleration is the rate of change of velocity.

Since the data is based on regular time differences, the velocity of the falling object can be studied by taking the differences in height values. That is, the first row of a difference table for the height data informs us about velocities.

And information about acceleration can be gleaned by looking at the changes of velocity, the second row of the difference table.

Galileo was seeing the same set of constant differences in the second row of his data tables for all the objects he tested.

Galileo’s Conclusions

Galileo saw that not only were objects of different masses accelerating in the same manner due to gravity, they were all accelerating at the same constant rate: acceleration due to gravity is a fixed constant value, independent of mass.

Displacement data from (ideal) falling motion gives constant second differences, provided the initial time measurements are taken at regular intervals.

Galileo then said that it follows that the change in height of a falling object is given by a quadratic expression,

\(at^2+bt+c\).

Whoa! Hold on! How does that follow?

This matches what we are told in physics class, but why does it follow? What is it about sequences that allows us to immediately know that the formula behind the data is quadratic?

It seems we need to understand the mathematics of sequences of numbers that have some structure within the differences between terms.

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