Physics 01, Lecture Today s Topics n Kinematics (Chap..1-.) n Position, Displacement (, and distance) n Time and Time Interval n Velocity (, and speed) n Acceleration *1-Dimension for today,,3-d later. n Expected from Preview n Definitions: Position, Displacement, distance n Velocity and speed (average and instantaneous) n Acceleration n... Announcement: Practicing Engineering Problem Solving(PrEPS) q PrEPS is a zero credit course (InterEGR 150, lab 13) administrated by the College of Engineering s Undergraduate Learning Center to help students with problem solving techniques. q There will be two (75 mininute) meetings each and every week (TR :30-7:5 pm in Wendt 305). Group discussions are facilitated by upper class engineering students. Our Physics 01 PrEPS facilitators are Genny Pfister ( glpfister@wisc.edu ) Neil Taylor (nptaylor@wisc.edu ) q If interested please email Kathy Prem kprem@wisc.edu for signing up The Big Picture q Physics 01 : Classical Mechanics q Mechanics: Physics about motion of objects Mechanics = Kinematics + Dynamics q Kinematics: to describe a motion Position, displacement (, and distance) Time, time interval Velocity (, and speed) Acceleration q Dynamics: to understand (what makes) a motion Forces and Newton s Laws (core of classical mechanics) Energy, Momentum, Angular Momentum,... Today: Basics Kinematics (1-Dimension) Kinematics: Position and Displacement q Position: a point in space (1,,or 3-D) e.g. x for 1-D, (x,y,z) for 3-D an object s position can be associated with a time (x 1,t 1 ), (x,t ), (x 3,t 3 ), (x,t ),... q Displacement: change in position of a single (point) object in a time interval. Δx x -x 1 in t 1 t. A displacement is always associated with two time points: a start time (t 1 ) and an end time (t ) A displacement is a vector (reducible to a signed number in 1-D) A displacement is defined regardless of the actual path in between t 1 and t. (x, t ) e.g. these three trips have the same displacement for t 1 and t. (x 1, t 1 ) 1
Displacement: an 1-D example q Joe s driving pattern 5 1 3 x 1 =0, x =5, x 3 =0, x =50, x 5 =0 km; (t 1 <t <t 3 <t <t 5 ) Ø Displacements: Δx 1 = x -x 1 = 5 km, Δx 3 = x 3 -x = -5 km, Δx 3 = x -x 3 =30 km, Δx 5 = x 5 -x =-50 km, Δx 1 = x -x 1 =50 km, Δx 15 = x 5 -x 1 =0 km! Δx 1 = Δx 1 + Δx 3 + Δx 3 =5+(-5)+30=50 km Δx 15 = Δx 1 +Δx 5 = 50 + (-50) = 0 km y x Quick Quiz 1 q If Joe s initial position is known and he has driven 10 km per odometer reading, can we determine where he is? Yes No Examples: Joe s possible driving patterns: Drove east straight for 10 km Drove south straight for 10 km Drove east for 0 km and west for another 0 Drove in circles for 10 km... *** Distance L 15 = Δx 1 + Δx 3 + Δx 3 + Δx 5 =110 km!= Δx 15 (assuming Joe did not change direction within each section) Generally, distance of travel can not be used to determine final position reliably(, but a displacement can) Distance of travel is not the same as displacement! Quick Quiz q If displacement for any time interval during Joe s trip is known, can we figure out the total distance he has driven? Yes No Ø Distance S = Σ Δx i S = dx = dx dt dt t t 1 Quick Quiz q If displacement for the whole time interval (t 1 -t ) during Joe s trip is known, can we figure out the total distance he has driven? Yes No (x, t ) Distance of travel is a non-fundamental but derivable quantity. (x 1, t 1 ) A (single) displacement tells the information about just the initial and final position.
Displacement and Travel Distance q Displacement is defined by initial and final positions only It is directional ( i.e. It is a vector ) It is path independent. q Travel distance is dependent on the entire travel path. i.e. It is so called path dependent It is a scalar (no direction, no sign). v Another related quantity is destination distance. (Geometric distance between initial and final positions) Ø Destination distance is simply the magnitude of displacement Ø It is a scalar and it is path independent. u Then what about a plain distance? u There is no uniform definition. u Loosely, if not specified in context, it refers to travel distance Velocity q Average velocity in time interval [t 1, t ] v ave Δx x x1 = t t 1 * Like displacement, velocity also has a direction (a vector!). Ø Instantaneous velocity at time t : Consider time interval [t, t + ], where 0 ( x( t + ) x( t)) Δx dx( t) v( t) = lim = t ( t t) t lim Δ 0 + Δ 0 dt q Two related quantities Instantaneous speed: S = v Average speed: S ave = distance of travel / time spent 1 D = x x1 t t After class question (ask your TA if necessary): is S ave = v? 1 unit: m/s ave AB x (m) 8 = = 1m/s 0 A AE BC AC Exercise: Reading x-t Graph B C = = 0 m/s = = 0.5 m/s 0 = = 0 m/s 5 0 CD 8 = = m/s 5 D DE 8 = = m/s 7 5 E EF F Basic Rules Straight line constant v Tangent of line v Negative tangent negative v Zero tangent (horizontal) v=0 Steeper tangent larger magnitude = = m/s 8 7 FG 0 = = 0.5 m/s 1 8 G Reading Graph: Instantaneous Velocity x (m) 8 A Δx v B =0 m/s B v A =Δx/ = 1.3 m/s C Basic Rules Tangent instantaneous v Negative tangent negative v Zero tangent (horizontal) v=0 Steeper tangent larger speed v C =- 0.m/s 8 10 1 1 t (s) 8 10 1 1 t (s) Find average velocity (respectively) in segment AB, BC, CD, DE, EF, FG, AC, AE Find the instantaneous velocities at A, B, and C, respectively 3
x (m) 8 Quiz 3: Reading x-t Graph Velocity and Speed q Average Velocity and Average Speed : Average Velocity = Displacement / (a vector quantity) Average Speed = Travel Distance / (a scalar quantity) Average Velocity Average Speed 8 10 1 1 t (s) q Instantaneous Velocity and Instantaneous Speed : Instantaneous Velocity = Displacement / ( 0) (vector) Instantaneous Speed: is defined as the magnitude of Instantaneous Velocity Quiz: How many zero instantaneous v s can you find in graph? A: none, B: one, C: two, D: three. Acceleration Exercise: Reading v-t Graph q Average acceleration in time interval [t 1, t ] Δv v v1 a ave = unit: m/s t t1 Ø Instantaneous acceleration at time t Considering time interval [t, t + ], when 0 a t) = ( v( t + ) v( t)) = ( t + ) t lim 0 0 ( lim Δv dv( t) dt v Like displacement and velocity, acceleration also has a direction. Tangent of the blue line Average acceleration between A and B Tangent of the green line Instantaneous acceleration at B
Position(x) Time(t) Graph Has Them All Summary of Concepts Displacement: change of position from t 1 t Velocity: rate of position change. Average: Δx/ Instantaneous: slope of x vs. t Acceleration: rate of velocity change. Average: Δv/ Instantaneous : slope of v vs. t (Note the similarity in math.) Summery: Kinematical Quantities to Describe a Motion Basic Quantities Displacement (Δx): change of position from t 1 t Velocity (v): rate of position change. Average: Δx/ Instantaneous: dx/dt Acceleration (a): rate of velocity change. Average: Δv/ Instantaneous : dv/dt v Note the mathematical similarity in v vs. x, and a vs. v Quick Quiz What can you say about t B t E? The car must be going in negative direction (-x) The car must be speeding up. The car must be slowing down None of above is necessarily true v speed s= v is not the same as v. Summery: Kinematical Quantities to Describe a Motion Basic Quantities Displacement (Δx): change of position from t 1 t Velocity (v): rate of position change. Average: Δx/ Instantaneous: dx/dt Acceleration (a): rate of velocity change. Average: Δv/ Instantaneous : dv/dt Quick Quiz What is happening at time t E? The car is tuning. The driver s foot must be moving from the gas paddle to the brake. The driver s foot must be moving from the brake to the gas paddle. The driver s foot is moving between the gas paddle and the brake. (either way possible). End-Of-Lecture Quiz 1 q If the average velocity of a car during a trip along a straight road is positive, is it possible for the instantaneous velocity at some time during the trip to be negative? A: Yes B: No v Note the mathematical similarity in v vs. x, and a vs. v v speed s= v is not the same as v. 5
End-Of-Lecture Quiz q If the velocity of some object is not zero, can its acceleration ever be zero? A: Yes B: No