Physics 207: Lecture 2, Pg 1 Lecture 2 Goals Goals: (Highlights of Chaps. 1 & )  Conduct order of magnitude calculations,  Determine units, scales, significant digits (in discussion or on your own)  Distinguish between Position & Displacement  Define Velocity (Average and Instantaneous), Speed  Define Acceleration  Understand algebraically, through vectors, and graphically the relationships between position, velocity and acceleration Dimensional Analysis  Perform Dimensional Analysis

Physics 207: Lecture 2, Pg 2 Reading Quiz Displacement, position, velocity & acceleration are the main quantities that we will discuss today. Which of these 4 quantities have the same units A. Velocity & position B. Velocity & acceleration C. Acceleration & displacement D. Position & displacement E. Position & acceleration

Physics 207: Lecture 2, Pg 3 Perspective Length/Time/Mass Distance Length (m) Radius of Visible Universe 1 x To Andromeda Galaxy 2 x To nearest star 4 x Earth to Sun 1.5 x Radius of Earth 6.4 x 10 6 Sears Tower 4.5 x 10 2 Football Field 1 x 10 2 Tall person 2 x 10 0 Thickness of paper 1 x Wavelength of blue light 4 x Diameter of hydrogen atom 1 x Diameter of proton 1 x Universal standard: The speed of light is defined to be exactly m/s and so one measures how far light travels in 1/ of a second

Physics 207: Lecture 2, Pg 4 Time IntervalTime (s) Interval Time (s) Age of Universe 5 x Age of Grand Canyon 3 x Avg age of college student 6.3 x 10 8 One year 3.2 x 10 7 One hour 3.6 x 10 3 Light travel from Earth to Moon 1.3 x 10 0 One cycle of guitar A string 2 x One cycle of FM radio wave 6 x One cycle of visible light 1 x Time for light to cross a proton 1 x World’s most accurate timepiece: Cesium fountain Atomic Clock Lose or gain one second in some 138 million years

Physics 207: Lecture 2, Pg 5 Mass Mass (kg) Stuff Mass (kg) Visible universe ~ Milky Way galaxy 7 x Sun 2 x Earth 6 x Boeing x 10 5 Car 1 x 10 3 Student 7 x 10 1 Dust particle 1 x Bacterium 1 x Proton 2 x Electron 9 x Neutrino <1 x

Physics 207: Lecture 2, Pg 6 Some Prefixes for Power of Ten PowerPrefix PowerPrefix Abbreviation 10 3 kilok 10 6 megaM 10 9 gigaG teraT petaP exaE attoa femtof picop nanon micro  millim

Physics 207: Lecture 2, Pg 7 Density l Every substance has a density, designated  = M/V Dimensions of density are, units (kg/m 3 ) Some examples, Substance  (10 3 kg/m 3 ) Gold19.3 Lead11.3 Aluminum 2.70 Water 1.00

Physics 207: Lecture 2, Pg 8 Atomic Density l In dealing with macroscopic numbers of atoms (and similar small particles) we often use a convenient quantity called Avogadro’s Number, N A = x atoms per mole l Commonly used mass units in regards to elements 1. Molar Mass = mass in grams of one mole of the substance (averaging over natural isotope occurrences) 2. Atomic Mass = mass in u (a.m.u.) of one atom of a substance. It is approximately the total number of protons and neutrons in one atom of that substance. 1u = x kg What is the mass of a single carbon (C 12 ) atom ? = 2 x g/atom

Physics 207: Lecture 2, Pg 9 Order of Magnitude Calculations / Estimates Question: If you were to eat one french fry per second, estimate how many years would it take you to eat a linear chain of trans-fat free french fries, placed end to end, that reach from the Earth to the moon? l Need to know something from your experience:  Average length of french fry: 3 inches or 8 cm, 0.08 m  Earth to moon distance: 250,000 miles  In meters: 1.6 x 2.5 X 10 5 km = 4 X 10 8 m  1 yr x 365 d/yr x 24 hr/d x 60 min/hr x 60 s/min = 3 x 10 7 sec

Physics 207: Lecture 2, Pg 10 Converting between different systems of units l Useful Conversion factors:  1 inch= 2.54 cm  1 m = 3.28 ft  1 mile= 5280 ft  1 mile = 1.61 km l Example: Convert miles per hour to meters per second:

Physics 207: Lecture 2, Pg 11 Home Exercise 1 Converting between different systems of units l When on travel in Europe you rent a small car which consumes 6 liters of gasoline per 100 km. What is the MPG of the car ? (There are 3.8 liters per gallon.)

Physics 207: Lecture 2, Pg 12 l This is a very important tool to check your work  Provides a reality check (if dimensional analysis fails then there is no sense in putting in numbers) l Example When working a problem you get an expression for distance d = v t 2 ( velocity · time 2 ) Quantity on left side d  L  length (also T  time and v  m/s  L / T) Quantity on right side = L / T x T 2 = L x T units and right units don’t match, so answer is nonsense l Left units and right units don’t match, so answer is nonsense Dimensional Analysis (reality check)

Physics 207: Lecture 2, Pg 13 Exercise 1 Dimensional Analysis l The force (F) to keep an object moving in a circle can be described in terms of:  velocity ( v, dimension L / T ) of the object  mass ( m, dimension M )  radius of the circle ( R, dimension L ) Which of the following formulas for F could be correct ? Note: Force has dimensions of ML/T 2 or kg-m / s 2 (a) (b) (c) F = mvR

Physics 207: Lecture 2, Pg 14 Exercise 1 Dimensional Analysis Which of the following formulas for F could be correct ? A.  B.  C.  F = mvR Note: Force has dimensions of ML/T 2 Velocity (, dimension L / T) Mass (m, dimension M) Radius of the circle (R, dimension L)

Physics 207: Lecture 2, Pg 15 Significant Figures l The number of digits that have merit in a measurement or calculation. l When writing a number, all non-zero digits are significant. l Zeros may or may not be significant.  those used to position the decimal point are not significant (unless followed by a decimal point)  those used to position powers of ten ordinals may or may not be significant. l In scientific notation all digits are significant l Examples: 21 sig fig 40ambiguous, could be 1 or 2 sig figs (use scientific notations) 4.0 x significant figures significant figures significant figures

Physics 207: Lecture 2, Pg 16 Significant Figures l When multiplying or dividing, the answer should have the same number of significant figures as the least accurate of the quantities in the calculation. l When adding or subtracting, the number of digits to the right of the decimal point should equal that of the term in the sum or difference that has the smallest number of digits to the right of the decimal point. l Examples:  2 x 3.1 = 6  4.0 x 10 1 / 2.04 x 10 2 = 1.6 X  2.4 – = 2.4 See:

Physics 207: Lecture 2, Pg 17 Moving between pictorial and graphical representations l Example: Initially I walk at a constant speed along a line from left to right, next smoothly slow down somewhat, then smoothly speed up, and, finally walk at the same constant speed. 1. Draw a pictorial representation of my motion by using a particle model showing my position at equal time increments. 2. Draw a graphical “xy” representation of my motion with time on the x-axis and position along the y-axis. Need to develop quantitative method(s) for algebraically describing: 1. Position 2. Rate of change in position (vs. time) 3. Rate of change in the change of position (vs. time)

Physics 207: Lecture 2, Pg 18 Tracking changes in position: VECTORS l Position l Displacement (change in position) l Velocity (change in position with time) l Acceleration (change in velocity with time) l Jerk (change in acceleration with time) Be careful, only vectors with identical units can be added/subtracted

Physics 207: Lecture 2, Pg 19 Motion in One-Dimension (Kinematics) Position l Position is usually measured and referenced to an origin:  At time= 0 seconds Pat is 10 meters to the right of the lamp  Origin  lamp  Positive direction  to the right of the lamp  Position vector : 10 meters Joe O -x +x 10 meters

Physics 207: Lecture 2, Pg 20 Pat xixi Displacement l One second later Pat is 15 meters to the right of the lamp l Displacement is just change in position   x = x f - x i 10 meters 15 meters O x f = x i +  x  x = x f - x i = 5 meters to the right !  t = t f - t i = 1 second Relating  x to  t yields velocity xfxf ΔxΔx

Physics 207: Lecture 2, Pg 21 Average Velocity Changes in position vs Changes in time Average velocity = net distance covered per total time, includes BOTH magnitude and direction Pat’s average velocity was 5 m/s to the right

Physics 207: Lecture 2, Pg 22 Average Speed l Speed, s, is usually just the magnitude of velocity.  The “how fast” without the direction. l However: Average speed references the total distance travelled Pat’s average speed was 5 m/s

Physics 207: Lecture 2, Pg 23 Representative examples of speed Speed (m/s) Speed (m/s) Speed of light 3x10 8 Electrons in a TV tube 10 7 Comets 10 6 Planet orbital speeds 10 5 Satellite orbital speeds 10 4 Mach Car 10 1 Walking 1 Centipede Motor proteins Molecular diffusion in liquids 10 -7

Physics 207: Lecture 2, Pg 24 Exercise 2 Average Velocity x (meters) t (seconds) What is the magnitude of the average velocity over the first 4 seconds ? (A) -1 m/s(D) not enough information to decide. (C) 1 m/s(B) 4 m/s

Physics 207: Lecture 2, Pg 25 Average Velocity Exercise 3 What is the average velocity in the last second (t = 3 to 4) ? A. 2 m/s B. 4 m/s C. 1 m/s D. 0 m/s x (meters) t (seconds)

Physics 207: Lecture 2, Pg 26 Average Speed Exercise 4 What is the average speed over the first 4 seconds ? 0 m to -2 m to 0 m to 4 m  8 meters total A. 2 m/s B. 4 m/s C. 1 m/s D. 0 m/s x (meters) t (seconds) turning point

Physics 207: Lecture 2, Pg 27 Instantaneous velocity Instantaneous velocity, velocity at a given instant If a position vs. time curve then just the slope at a point x t

Physics 207: Lecture 2, Pg 28 What is the instantaneous velocity at the fourth second? (A) 4 m/s(D) not enough information to decide. (C) 1 m/s(B) 0 m/s Exercise 5 Instantaneous Velocity x (meters) t (seconds) Instantaneous velocity, velocity at a given instant

Physics 207: Lecture 2, Pg 29 What is the instantaneous velocity at the fourth second? (A) 4 m/s(D) not enough information to decide. (C) 1 m/s (B) 0 m/s x (meters) t (seconds) Exercise 5 Instantaneous Velocity

Physics 207: Lecture 2, Pg 30 Key point: position  velocity (1) If the position x(t) is known as a function of time, then we can find instantaneous or average velocity v vxvx t x t l (2) “Area” under the v(t) curve yields the displacement l A special case: If the velocity is a constant, then x(Δt)=v Δt + x 0

Physics 207: Lecture 2, Pg 31 Motion in Two-Dimensions (Kinematics) Position / Displacement l Amy has a different plan (top view):  At time= 0 seconds Amy is 10 meters to the right of the lamp (East)  origin = lamp  positive x-direction = east of the lamp  position y-direction = north of the lamp 10 meters Amy O -x +x 10 meters N

Physics 207: Lecture 2, Pg 32 “2D” Position, Displacement time (sec) position 1,0 2,0 3,0 4,0 5,0 6,0 (x,y meters) origin position vectors x y

Physics 207: Lecture 2, Pg 33 “2D” Position, Displacement time (sec) position 1,0 2,0 3,0 4,0 5,0 6,0 (x,y meters) origin position vectors x y displacement vectors

Physics 207: Lecture 2, Pg 34 Position, Displacement, Velocity time (sec) origin position vectors displacement vectors velocity vectors and, this case, there is no change in velocity x y

Physics 207: Lecture 2, Pg 35 Acceleration l A particle’s motion often involve acceleration  The magnitude of the velocity vector may change  The direction of the velocity vector may change (true even if the magnitude remains constant)  Both may change simultaneously l Now a “particle” with smoothly decreasing speed v0v0 v1v1 v1v1 v1v1 v3v3 v5v5 v2v2 v4v4 v0v0

Physics 207: Lecture 2, Pg 36 Constant Acceleration l Changes in a particle’s motion often involve acceleration  The magnitude of the velocity vector may change l A particle with smoothly decreasing speed: v1v1v1v1 v0v0v0v0 v3v3v3v3 v5v5v5v5 v2v2v2v2 v4v4v4v4 a a a a a a v t 0 v(  t)=v 0 + a  t a  t = area under curve =  v (an integral) a t tt

Physics 207: Lecture 2, Pg 37 Instantaneous Acceleration l The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero l Position, velocity & acceleration are all vectors and must be kept separate (i.e., they cannot be added to one another)

Physics 207: Lecture 2, Pg 38 Assignment l Reading for Tuesday’s class » Finish Chapter 2 & all of 3 (vectors) » HW 0 due soon, HW1 is available