Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Sections 807, 808, 809 Lecture 3

Integrals Indefinite integral: n – any number except -1 Definite integral:

Gottfried Leibniz These are Leibniz’ notations: Integral sign as an elongated S from “Summa” and d as a differential (infinitely small increment)

Leibniz-Newton calculus priority dispute

Galileo Galilei

“Father of modern science” Was the first to apply a scientific method: Put forward a hypothesis, verify it by experiment, describe it with a mathematical model Insisted that language of mathematics should describe the laws of nature and experiments should prove it. No place for arguments based on beauty, religion etc. Stephen Hawking: Galileo probably bears more of the responsibility for the birth of modern science than anybody else. Albert Einstein

Achievements in physics Verified that free-fall acceleration is independent on masses of bodies. This fact inspired Einstein’s General Relativity. Formulated the Principle of Relativity, which laid the framework for Newton’s laws and inspired Einstein’s Special Relativity. Proposed the Principle of Inertia, which was used (borrowed?) by Newton as his First Law. Found that the period of a pendulum is independent on its amplitude. He discovered it by observing swings of a bronze chandelier in the Cathedral of Pisa and using his pulse to measure the time!

a v a = g = const for all bodies independently on their masses Galileo Galilei ( ), the basic law of motion

Basilica di Santa Croce, Florence

Burial place of famous Florentines. Among them: Galileo Michelangelo Machiavelli Dante Rossini Fermi Marconi

Free fall

Falling with air resistance

Galileo's “Law of Falling Bodies” distance (S) is proportional to time (T) squared

Motion in One Dimension (Chapter 2) We consider a particle - as time goes, the position of the particle changes

Velocity is the rate at which the position changes with time Average velocity:

Acceleration is the rate at which the velocity changes with time Average acceleration

If a=a c =Const:

A “police car” problem x=0 x1x1 x2x2 V 3 =20m/s a=0 a p =kt x 2 – x 1 = 3.5 km V 1 =30m/sV 2 =40m/s 1. What was your acceleration before you meet the second police car? 2. How much time did you travel from x 1 to x 2 ? 3. Find x 1 4. At which distance the police car catches you? 5.Convert the velocity from m/s to mph a=const V(t=0)=0

Have a great day! Reading: Chapter 2