Why units matter! Flight 173 ran out of fuel in flight…. You may ask, “So, how does a jet run out of fuel at 26000 ft ???" 1.  A maintenance worker found that the fuel gauge did not work on ground inspection.  He incorrectly assured the pilot that the plane was certified to fly without a functioning fuel gauge if the crew checked the fuel tank levels. 2.  Crew members measured the 2 fuel tank levels at 62 cm and 64 cm.  This corresponded to 3758 L and 3924 L for a total of 7682 L according to the plane's manual. 3.  The ground crew knew that the flight required 22,300 kg of fuel.  The problem they faced was with 7,682 L of fuel on the plane, how many more liters were needed to total 22,300 kg of fuel? 4.  One crew member informed the other that the "conversion factor" (being the fuel density) was 1.77.  THE CRUCIAL FAULT BEING THAT NO ONE EVER INQUIRED ABOUT THE UNITS OF THE CONVERSION FACTOR.  So it was calculated that the plane needed an additional 4,917 L of fuel for the flight. Alas, that was too little. 1

Lecture 2 Goals: (Highlights of Chaps. 1 & 2.1-2.4) 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 Perform Dimensional Analysis 1

For next class: Finish reading Ch. 2, read Chapter 3 (Vectors) Lecture 2 Reading Assignment: For next class: Finish reading Ch. 2, read Chapter 3 (Vectors) 1

Length Distance Length (m) Radius of Visible Universe 1 x 1026 To Andromeda Galaxy 2 x 1022 To nearest star 4 x 1016 Earth to Sun 1.5 x 1011 Radius of Earth 6.4 x 106 Sears Tower 4.5 x 102 Football Field 1 x 102 Tall person 2 x 100 Thickness of paper 1 x 10-4 Wavelength of blue light 4 x 10-7 Diameter of hydrogen atom 1 x 10-10 Diameter of proton 1 x 10-15 See http://micro.magnet.fsu.edu/primer/java/scienceopticsu

Time Interval Time (s) Age of Universe 5 x 1017 Age of Grand Canyon 3 x 1014 Avg age of college student 6.3 x 108 One year 3.2 x 107 One hour 3.6 x 103 Light travel from Earth to Moon 1.3 x 100 One cycle of guitar A string 2 x 10-3 One cycle of FM radio wave 6 x 10-8 One cycle of visible light 1 x 10-15 Time for light to cross a proton 1 x 10-24

Mass Stuff Mass (kg) Visible universe ~ 1052 Milky Way galaxy 7 x 1041 Sun 2 x 1030 Earth 6 x 1024 Boeing 747 4 x 105 Car 1 x 103 Student 7 x 101 Dust particle 1 x 10-9 Bacterium 1 x 10-15 Proton 2 x 10-27 Electron 9 x 10-31 Neutrino <1 x 10-36

Some Prefixes for Power of Ten Power Prefix Abbreviation 10-18 atto a 10-15 femto f 10-12 pico p 10-9 nano n 10-6 micro m 10-3 milli m 103 kilo k 106 mega M 109 giga G 1012 tera T 1015 peta P 1018 exa E

Density Every substance has a density, designated  = M/V Dimensions of density are, units (kg/m3) Some examples, Substance  (103 kg/m3) Gold 19.3 Lead 11.3 Aluminum 2.70 Water 1.00

What is the mass of a single carbon (C12) atom ? Atomic Density In dealing with macroscopic numbers of atoms (and similar small particles) we often use a convenient quantity called Avogadro’s Number, NA = 6.023 x 1023 atoms per mole 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 = 1.660 538 7 x 10-27 kg What is the mass of a single carbon (C12) atom ? = 2 x 10-23 g/atom

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? 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 105 km = 4 X 108 m 1 yr x 365 d/yr x 24 hr/d x 60 min/hr x 60 s/min = 3 x 107 sec

Converting between different systems of units Useful Conversion factors: 1 inch = 2.54 cm 1 m = 3.28 ft 1 mile = 5280 ft 1 mile = 1.61 km Example: Convert miles per hour to meters per second:

Home Exercise 1 Converting between different systems of units 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.)

Dimensional Analysis (reality check) 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) Example When working a problem you get an expression for distance d = v t 2 ( velocity · time2 ) Quantity on left side d  L  length (also T  time and v  m/s  L / T) Quantity on right side = L / T x T2 = L x T Left units and right units don’t match, so answer is nonsense

Exercise 1 Dimensional Analysis 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/T2 or kg-m / s2 F = mvR (a) (b) (c)

Note: Force has dimensions of ML/T2 Exercise 1 Dimensional Analysis Which of the following formulas for F could be correct ? Note: Force has dimensions of ML/T2 Velocity (n, dimension L / T) Mass (m, dimension M) Radius of the circle (R, dimension L) F = mvR 

Significant Figures The number of digits that have merit in a measurement or calculation. When writing a number, all non-zero digits are significant. 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. In scientific notation all digits are significant Examples: 2 1 sig fig 40 ambiguous, could be 1 or 2 sig figs (use scientific notations) 4.0 x 101 2 significant figures 0.0031 2 significant figures 3.03 3 significant figures

Significant Figures When multiplying or dividing, the answer should have the same number of significant figures as the least accurate of the quantities in the calculation. 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. Examples: 2 x 3.1 = 6 4.0 x 101 / 2.04 x 102 = 1.6 X 10-1 2.4 – 0.0023 = 2.4 See: http://lectureonline.cl.msu.edu/~mmp/applist/sigfig/sig.htm

Moving between pictorial and graphical representations 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. 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: Position Rate of change in position (vs. time) Rate of change in the change of position (vs. time)

Tracking changes in position: VECTORS Displacement (change in position) Velocity (change in position with time) Acceleration

Motion in One-Dimension (Kinematics) Position / Displacement 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 2

Position / Displacement One second later Pat is 15 meters to the right of the lamp Displacement is just change in position x = xf - xi 10 meters 15 meters Δx Pat xi xf O xf = xi + x x = xf - xi = 5 meters to the right ! t = tf - ti = 1 second 2

Average Speed & Velocity Changes in position vs Changes in time Average velocity = net distance covered per total time, includes BOTH magnitude and direction Speed, s, is usually just the magnitude of velocity. The “how fast” without the direction. Average speed references the total distance travelled Active Figure 1 http://www.phy.ntnu.edu.tw/ntnujava/main.php?t=282

Representative examples of speed Speed (m/s) Speed of light 3x108 Electrons in a TV tube 107 Comets 106 Planet orbital speeds 105 Satellite orbital speeds 104 Mach 3 103 Car 101 Walking 1 Centipede 10-2 Motor proteins 10-6 Molecular diffusion in liquids 10-7

Instantaneous velocity Changes in position vs Changes in time Instantaneous velocity, velocity at a given instant If a position vs. time curve then just the slope at a point x t http://www.phy.ntnu.edu.tw/ntnujava/main.php?t=230

Exercise 2 Average Velocity x (meters) 6 4 2 t (seconds) -2 1 2 3 4 What is the magnitude of the average velocity over the first 4 seconds ? (A) -1 m/s (B) 4 m/s (C) 1 m/s (D) not enough information to decide.

Average Velocity Exercise 3 What is the average velocity in the last second (t = 3 to 4) ? x (meters) 6 4 2 -2 t (seconds) 1 2 3 4 2 m/s 4 m/s 1 m/s 0 m/s

Exercise 4 Instantaneous Velocity x (meters) 6 4 2 -2 t (seconds) 1 2 3 4 What is the instantaneous velocity at the fourth second? (A) 4 m/s (B) 0 m/s (C) 1 m/s (D) not enough information to decide.

Average Speed Exercise 5 What is the average speed over the first 4 seconds ? 0 m to -2 m to 0 m to 4 m  8 meters total x (meters) t (seconds) 2 6 -2 4 1 3 turning point 2 m/s 4 m/s 1 m/s 0 m/s

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 x t vx t (2) “Area” under the v(t) curve yields the change in position A special case: If the velocity is a constant, then x(Δt)=v Δt + x0

Exercise 6, (and some things are easier than they appear) A marathon runner runs at a steady 15 km/hr. When the runner is 7.5 km from the finish, a bird begins flying from the runner to the finish at 30 km/hr. When the bird reaches the finish line, it turns around and flies back to the runner, and then turns around again, repeating these back-and-forth trips until the runner reaches the finish line. How many kilometers does the bird fly? Hint: Don’t focus on the bird. How many kilometers does the runner travel and how long does that take? A. 10 km B. 15 km C. 20 km D. 30 km

Motion in Two-Dimensions (Kinematics) Position / Displacement 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 N 10 meters Amy O -x +x 10 meters 2

“2D” Position, Displacement y time (sec) 1 2 3 4 5 6 position -2,2 -1,2 0,2 1,2 2,2 3,2 (x,y meters) x position vectors origin Continued on Monday….

Reading for Monday’s class on 9/13 Assignment Recap Reading for Monday’s class on 9/13 Finish Chapter 2 & all of 3 (vectors) 27