Lecture 2

1. Lecture 2 Goals: (Highlights of Chaps. 1 & 2.1-2.4) • Units and scales, order of magnitude calculations, significant digits (on your own for the most part) • 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

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

3. 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 proton1 x 10-15 See http://micro.magnet.fsu.edu/primer/java/scienceopticsu

4. Time IntervalTime (s) Age of Universe 5 x 1017 Age of Grand Canyon 3 x 1014 Avg age of college student 6.3 x 108 One year3.2 x 107 One hour3.6 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

5. Order of Magnitude Calculations / EstimatesQuestion: 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

6. Dimensional Analysis (reality check) • This is a very important tool to check your work • Provides a reality check (if dimensional analysis fails then no sense in putting in the 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

7. Exercise 1Dimensional 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 Fcould be correct ? Note: Force has dimensions of ML/T2 orkg-m / s2 F = mvR (a) (c) (b)

8. 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)

9. Tracking changes in position: VECTORS • Position • Displacement (change in position) • Velocity (change in position with time) • Acceleration

10. Joe +x -x 10 meters O Motion in One-Dimension (Kinematics) Position / Displacement • Position is usually measured and referenced to an origin: • At time= 0 seconds Joe is 10 meters to the right of the lamp • origin = lamp • positive direction = to the right of the lamp • position vector : 10 meters

11. 15 meters xi Joe Position / Displacement • One second later Joe is 15 meters to the right of the lamp • Displacement is just change in position. • x = xf - xi 10 meters Δx xf O xf = xi + x x = xf - xi = 5 metersto the right ! t = tf - ti= 1 second

12. Speed & VelocityChanges in positionvsChanges in time • Average velocity = net distance covered per total time, • 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

13. 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

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

15. Exercise 2Average Velocity x (meters) 6 4 2 0 t (seconds) 1 2 3 4 -2 What is 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.

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

17. x (meters) 6 4 2 -2 t (seconds) 1 2 3 4 Exercise 4Instantaneous Velocity • Instantaneous velocity, velocity at a given instant 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.

18. Average Speed Exercise 5 What is the average speed over the first 4 seconds ? Here we want the ratio: total distance travelled / time(Could have asked “what was the speed in the first 4 seconds?”) • 2 m/s • 4 m/s • 1 m/s • 0 m/s x (meters) 6 4 2 -2 t (seconds) 1 2 3 4 turning point

19. Position, Displacement, Velocity, Acceleration y time (sec) 1 2 3 4 5 6 position 1 2 3 4 5 6 (meters) x position vectors origin displacement vectors velocity vectors 1 2 3 4 5 6 and, this case, there is no change in velocity

20. v0 v1 v1 v1 v3 v5 v2 v4 v0 Acceleration • Changes in 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 • Here, a “particle” with smoothly decreasing speed

21. v1 v0 v3 v5 v2 v4 a a a a a a a t t 0 Constant Acceleration • Changes in a particle’s motion often involve acceleration • The magnitude of the velocity vector may change • A particle with smoothly decreasing speed: Dt v(Dt)=v0 + a Dt v a Dt = area under curve = Dv (an integral)

22. Instantaneous Acceleration • The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero

23. Position, velocity & acceleration • All are vectors! • Cannot be used interchangeably (different units!) (e.g., position vectors cannot be added directly to velocity vectors) • But the directions can be determined • “Change in the position” vector vs. time gives the direction of the velocity vector • “Change in the velocity” vector vs. time gives the direction of the acceleration vector • Given x(t)  vx(t)  ax (t) • Given ax (t)  vx (t)  x(t)

24. And given a constant acceleration we can integrate to get explicit v and a x t vx t ax t

25. A “biology” experiment • Hypothesis: Older people have slower reaction times • Distance accentuates the impact of time differences • Equipment: Ruler and three volunteers • Older student • Younger student • Record keeper • Multiple trials to minimize statistics

26. Assignment Recap • Reading for Tuesday’s class • Finish Chapter 2 & all of 3 (vectors) • And first assignment is due this Wednesday