Physics Part 1 MECHANICS Topic III. Motion in Two Dimension (Projectile Motion) INCOMPLETE W. Pezzaglia Updated: 2012Aug27

2 II. Motion in 2D Vectors Displacement and Velocity Acceleration

A. Vectors History of Discovery Definition & Representation 3 A. Vectors History of Discovery Definition & Representation Vector Algebra

4 1. History of Vectors Aristotle probably knew that forces in different directions add by a parallelogram rule Leonardo da Vinci stated this principle Father Lami’s rule (1687)

1a. The Discovery of 3D vectors 5 which happened to be a Monday, and a Council day of the Royal Irish Academy – I was walking in to attend and preside, …, along the Royal Canal, … an under-current of thought was going on in my mind, which at last gave a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, … I could not resist the impulse … to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formulae…. Sir William Rowan Hamilton 1805-1865

The famous equation written at bridge 6

1b. Grassmann Algebra 7 Hermann Grassmann (1809-1877) Inventor of “Linear Algebra” 1844 publishes massive work (which nobody understands) [1 year after quaternions!]

1c. Gibbs Vectors 8 1881 Inventor of the vector system we now use. Oliver Heaviside used these vectors to reformulate Maxwell’s equations “... a sort of hermaphrodite monster, compounded of the notations of Hamilton and Grassmann” -Tait

2. Definition & Representation 9 (a). Geometric Vector A “directed line segment” Magnitude: the length Origin: the initial point Terminus: the “head” or arrow

Rest of lecture, look under the “topics” menu item for handwritten lecture notes.

b. Cartesian Coordinates 11 Vector defined by (x,y,z) coordinates A= axi +ayj+azk Or: A= (ax,ay,az)

Displacement vs Distance 12 Displacement vs Distance Position: a point on the graph Displacement: a change in position If you return to the same place (e.g. a circle) then the displacement is ZERO. Distance traveled is NOT zero however.

C. Projectile Motion Path of a projectile 13 Old view was that projectile travelled in an arc until it ran out of impetus and then it fell straight down. Galileo shows the natural path is a parabola (which is a combination of constant speed in the horizontal motion with constant acceleration in the vertical)

2. Projectile Motion 14

15 3. Simultaneous Fall Simultaneous Fall: Galileo shows a bullet fired horizontal will hit ground at same time as bullet dropped. (why?)

3b Simultaneous Fall 16 Both balls fall in the vertical direction at the same acceleration. Their paths only differ because of the constant horizontal velocity

References & Links Simultaneous Fall: 20:35-21:25 of Mechanical Universe #6 Newton’s Laws http://video.google.com/videoplay?docid=8080785615223230317&ei=en And http://www.google.com/url?sa=t&rct=j&q=galileo%20simultaneous%20fall&source=video&cd=2&ved=0CD0QtwIwAQ&url=http%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3Dp-A-T4Wfu6A&ei=O5nyT5x8w7DbBabPzboI&usg=AFQjCNGtIaeKD63FZFXvASlvHMY3Lh7bOA&cad=rja Descartes, R. La Géométrie. Livre Premier: Des problèmes qu'on peut construire sans y employer que des cercles et des lignes droites (Book one: Problems whose construction requires only circles and straight lines). (French) Galileo, Two New Sciences (1638). Chapter “Third Day” discusses motion and acceleration. Origin of terms abscissa and ordinate, see http://www.pballew.net/arithme1.html#abscissa Video: Mechanical Universe #2: Law of Falling bodies: http://video.google.com/videoplay?docid=3641917188010584794# PHET: Projectile Motion: http://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.html SAT Physics: http://www.sparknotes.com/testprep/books/sat2/physics/chapter5.rhtml Good Java Demo: http://www.jimloy.com/cindy/galilean.htm Galileo interview: http://www.youtube.com/watch?v=YgtOXNcOW1s Galileo: http://www.youtube.com/watch?v=mNMZlcyIj-8

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