We investigated what properties would determine if an object would float and arrived at the conclusion that if an object has a density r_obj less than the density of water r_water that it would float. Using Archimedes result that an object is buoyed up by a force equal to the weight of water it displaces and some algebra, we arrived at the formula

V_submerged / V_obj = r_obj / r_water

or that the ratio of the volume of the object that is under water to the volume of the object (a.k.a. the percentage of the object that is under water) is equal to the ratio of the object's density to the density of water (assuming that it is floating in water).

We considered that the shape of the object played some role. For example, we could can a boat from steel (which is denser than water) but then the air (and whatever else)in the boat's hull would have to be included in our density calculation. And if there were a hole in the boat, ....

We went on to discuss dead bodies and the Dead Sea -- that processes in dead bodies (presumably some decay process) change their density, and that the water in the Dead Sea in denser than normal a greater percentage of live bodies will there.

We initiated our consideration of Galileo, mentioning his observation of the moons of Jupiter, his furthering the theories of Copernicus (the heliocentric theory), as well as his work on bodies of different masses falling at the same rate. We discussed that he was opposing the authority of Aristotle with some of his work and the authority of the pope with another part of his work.

We started a discussion about Galileo's work on falling bodies and about acceleration.  We noted that we had already used the quantity g=9.8 m/s² ("the acceleration due to gravity") as a conversion factor between mass and weight, but we have not tried to understand it as an acceleration.

We introduced another example of an acceleration to further the conversation, namely a car that could go from 0 to 50 (miles per hour) in 6 seconds. This brought out that concept that acceleration is a rate of change of velocity; in it we see the change in the velocity from 0 to 50 and that this takes place at a certain rate or within a certain time frame.

We also noted that velocity was a vector (though I don't think we used that term), indicating that it has both what we call a magnitude (in this case the speed) and a direction. So 50 mph is a speed, but 50 mph North is a velocity. We noted that an acceleration then could be a change in speed or a change in direction. This anticipated an idea that will arise when we get to Newton, that the moon revolving around the earth and the apple falling to the ground are essentially the same, though the former is mainly a change in direction and the latter a change in speed.

We examined a conversion to put our two examples of accelerations into the same units.

(50 miles/hour)/(6 seconds)*(1 hour/60 minutes)*(1 minute/60 seconds)* (5 kilometers/3.1 miles)*(1000 meters/kilometer) = 3.73 m/s²

We considered the distance covered by the car in 6 seconds, arguing that it was equal to the distance that a car going 25 miles per hour would go in 6 seconds, where 25 miles per hour is the average speed during the time. We continued by noting that if the car starts from rest and and has a constant acceleration of a, then the velocity after time t is a t (for example, the car is going (50/6) mph in 1 sec., (2*50/6) mph in 2 sec., (3*50/6) mph in 3 sec., etc.)

Furthermore, if you start from rest and have constant acceleration, then the average speed up to time t is one half of the speed at time t (i.e., a*t/2). Moreover, the distance covered up to time t is then a t²/2.