Michael Fowler, UVa
1. State in your own words what the Greeks meant by natural motion and violent motion. What was the connection between natural motion and the four elements (and what were they?)
2. Draw a velocity against time graph for the ball rolling down the ramp, assuming it is accelerating uniformly and is going one meter per second after two seconds. What is its average speed during the first two seconds? How far does it travel in that two second period? What is its average speed during the first four seconds? (Assume the ramp is long enough!) How far does it travel during that four seconds?
3. Galileo states (TNS, page 151): "The distances traversed, during equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity."
In other words, if you count the distance fallen from rest in the first second as your unit of length, then in the second second the body falls a further 3 units, in the third second it falls another 5, in the fourth 7, etc.
Draw a velocity time graph, find from it the average speed in each of these successive one-second intervals, and confirm Galileo's statement.
4. Suppose in rolling the ball down the ramp you gave it an initial push so that it started at two meters per second, but gained speed at the same rate as in question 2. Draw a new velocity time graph, and find from the graph how far the ball rolls in the first second.
5. Suppose now you'd given the ball an initial push up the ramp, so it begins with an upward velocity of one meter per second. Draw another velocity time graph, and use the graph to figure out how far the ball has traveled after four seconds.