We considered that part of Galileo's legacy was to change the focus from objects' velocities (e.g. the terminal velocity that results when the air resistance force and gravitational force balance) to objects' accelerations. Starting with the notion of acceleration as "change in velocity over time", we considered that velocity was speed in a particular direction, and thus an object is accelerating if either its speed or its direction is changing.

We discussed the mass on an inclined plane experiment determining possible variables such as the mass of the object (which ideally in the absence of air resistance and friction should not affect the acceleration) and the angle of incline (which ideally changes the acceleration as g*sin(θ)).  Next we considered the displacement of the object as a function of time -- or in Galileo's case as a function of the amount of water issued from his water clock.  

Playing concepts like "final velocity", "average velocity", etc. against each other, we arrived at the following expression

x(t) = x₀ + v₀t + a t²/2

for the position as a function of time for an object experiencing constant acceleration, where x₀ is the initial position, v₀ is the initial velocity and a is the acceleration. We noted that this result had the form of a polynomial of order 2, a.k.a. a quadratic and when x₀=0 and v₀=0 it also has the form of a power-law (which we distinguished from an exponential behavior).

Finally, we considered another kind of experiment -- the "thought" experiment. Galileo could argue against some of the prevailing ideas about masses and speeds by considering dropping a "composite" object. Based on their arguments Galileo's opponents would be led to conclude that the composite object would fall at one speed if thought of as a single object and at a different speed if thought of as comprised of two or more objects.