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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 Physics 207: Lecture 2, Pg 1 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 2 Perspective Length/Time/Mass Distance Radius of Visible Universe To Andromeda Galaxy To nearest star Earth to Sun Radius of Earth Sears Tower Football Field Tall person Thickness of paper Wavelength of blue light Diameter of hydrogen atom Diameter of proton Length (m) 1 x 1026 2 x 1022 4 x 1016 1.5 x 1011 6.4 x 106 4.5 x 102 1 x 102 2 x 100 1 x 10-4 4 x 10-7 1 x 10-10 1 x 10-15 Universal standard: The speed of light is defined to be exactly 299 792 458 m/s and so one measures how far light travels in 1/299 792 458 of a second Physics 207: Lecture 2, Pg 3 Time Interval Age of Universe Age of Grand Canyon Avg age of college student One year One hour Light travel from Earth to Moon One cycle of guitar A string One cycle of FM radio wave One cycle of visible light Time for light to cross a proton Time (s) 5 x 1017 3 x 1014 6.3 x 108 3.2 x 107 3.6 x 103 1.3 x 100 2 x 10-3 6 x 10-8 1 x 10-15 1 x 10-24 World’s most accurate timepiece: Cesium fountain Atomic Clock Lose or gain one second in some 138 million years Physics 207: Lecture 2, Pg 4 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 4 108 m 10 ff 0 . 5 10 (to moon) 2 8 10 m 9 5 10 s 10 0.5 10 sec. 200 years 7 3 10 s/yr Physics 207: Lecture 2, Pg 9 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 Physics 207: Lecture 2, Pg 12 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 (a) F = mvR (b) v F m R 2 (c) kg-m / s2 mv2 F R Physics 207: Lecture 2, Pg 13 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) A. F = mvR Mass (m, dimension M) B. v F m R C. 2 mv F R 2 Radius of the circle (R, dimension L) Physics 207: Lecture 2, Pg 14 Tracking changes in position: VECTORS Position Displacement (change in position) Velocity (change in position with time) Acceleration (change in velocity with time) Jerk (change in acceleration with time) Be careful, only vectors with identical units can be added/subtracted Physics 207: Lecture 2, Pg 18 Motion in One-Dimension (Kinematics) Position Position is usually measured and referenced to an origin: 10 meters 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 : -x +x 10 meters O Joe Physics 207: Lecture 2, Pg 19 Displacement One second later Pat is 15 meters to the right of the lamp Displacement is just change in position x = xf - xi 15 meters 10 meters xi Δx xf Pat O xf = xi + x x = xf - xi = 5 meters to the right ! t = tf - ti = 1 second Relating x to t yields velocity Physics 207: Lecture 2, Pg 20 Average Velocity Changes in position vs Changes in time • Average velocity = net distance covered per total time, includes BOTH magnitude and direction x(net displaceme nt ) vaverage velocity t ( total time ) x(5 m to the right ) vaverage velocity t (1 sec) • Pat’s average velocity was 5 m/s to the right Physics 207: Lecture 2, Pg 21 Average Speed Speed, s, is usually just the magnitude of velocity. The “how fast” without the direction. However: Average speed references the total distance travelled distance taken along path saverage speed t ( total time ) • Pat’s average speed was 5 m/s Physics 207: Lecture 2, Pg 22 Exercise 2 Average Velocity x (meters) 6 4 2 0 -2 1 2 3 4 t (seconds) 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. Physics 207: Lecture 2, Pg 24 Average Velocity Exercise 3 What is the average velocity in the last second (t = 3 to 4) ? x (meters) 6 4 2 -2 A. B. C. D. 1 2 3 4 t (seconds) 2 m/s 4 m/s 1 m/s 0 m/s Physics 207: Lecture 2, Pg 25 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 x (meters) 6 4 2 A. B. C. D. 2 m/s 4 m/s 1 m/s 0 m/s -2 1 2 3 4 t (seconds) turning point Physics 207: Lecture 2, Pg 26 Instantaneous velocity • Instantaneous velocity, velocity at a given instant x(displaceme nt ) dx vvelocity lim t 0 t ( time ) dt • If a position vs. time curve then just the slope at a point x vinstantane ous x f xi lim t 0 t t f i t Physics 207: Lecture 2, Pg 27 Exercise 5 Instantaneous Velocity • x (meters) 6 dx vvelocity dt 4 2 -2 Instantaneous velocity, velocity at a given instant 1 2 3 4 t (seconds) 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. Physics 207: Lecture 2, Pg 28 Exercise 5 Instantaneous Velocity x (meters) 6 4 2 -2 1 2 3 4 t (seconds) 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. Physics 207: Lecture 2, Pg 29 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 x(t ) dx vx dt t1 x dt vx (t ) x t vx t0 (2) “Area” under the v(t) curve yields the displacementt A special case: If the velocity is a constant, then t1 x vx dt area of rectangle t0 x(Δt)=v Δt + x0 Physics 207: Lecture 2, Pg 30 Acceleration 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 Now a “particle” with smoothly decreasing speed v1 v0 v v1 v0 v v1 v2 v v v v1 v0 v5 v4 v3 v v aavg v t Physics 207: Lecture 2, Pg 35 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: aavg a v t v0 v1 v2 v3 v4 v5 a a a a a a t t v v(t)=v0 + a t 0 a t = area under curve = v (an integral) t Physics 207: Lecture 2, Pg 36 Instantaneous Acceleration The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero 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 37 Assignment Reading for Tuesday’s class » Finish Chapter 2 & all of 3 (vectors) » HW 0 due soon, HW1 is available Physics 207: Lecture 2, Pg 38