1Chapter 2 - Motion along a straight lineI. Position and displacementII. VelocityIII. AccelerationIV. Motion in one dimension with constant accelerationV. Free fallMECHANICS  KinematicsParticle: point-like object that has a mass but infinitesimal size.I. Position and displacementPosition: Defined in terms of a frame of reference: x or y axis in 1D.- The object’s position is its location with respect to the frame of reference.The smooth curve is a guess as to what happened between the data points.Position-Time graph: shows the motion of the particle (car).2I. Position and displacementDisplacement: Change from position x1to x2  ∆x = x2-x1(2.1)during a time interval. - Vector quantity: Magnitude (absolute value) and direction (sign).- Coordinate (position) ≠ Displacement  x ≠ ∆xtx∆x = 0 x1=x2Only the initial and final coordinates influence the displacement  many different motions between x1and x2 give the same displacement.tx∆x >0x1x2Coordinate systemDistance: length of a path followed by a particle.Displacement ≠ DistanceExample: round trip house-work-house  distance traveled = 10 kmdisplacement = 0 - Scalar quantity- Vector quantities need both magnitude (size or numerical value) and direction to completely describe them.- We will use + and – signs to indicate vector directions.- Scalar quantities are completely described by magnitude only.Review:3II. Velocity)2.2(ttxx∆t∆xv1212avg−−==Average velocity: Ratio of the displacement ∆x that occurs during a particular time interval ∆t to that interval.Motion along x-axis-Vector quantity  indicates not just how fast an object is moving but also in which direction it is moving.- SI Units: m/s- Dimensions: Length/Time [L]/[T]- The slope of a straight line connecting 2 points on an x-versus-t plot is equal to the average velocityduring that time interval.Average speed: Total distance covered in a time interval.)3.2(∆tdistanceTotalSavg=Example: A person drives 4 mi at 30 mi/h and 4 mi and 50 mi/h  Is the average speed >,<,= 40 mi/h ?<40 mi/ht1= 4 mi/(30 mi/h)=0.13 h ; t2= 4 mi/(50 mi/h)=0.08 h  ttot= 0.213 h  Savg= 8 mi/0.213h = 37.5mi/hSavg≠ magnitude VavgSavgalways >0Scalar quantitySame units as velocity4Instantaneous velocity: How fast a particle is moving at a given instant.)4.2(lim0dtdxtxvtx=∆∆=→∆-Vector quantity- The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero.- The instantaneous velocity indicates what is happening at every point of time.- Can be positive, negative, or zero.- The instantaneous velocity is the slope of the line tangent to the x vs. t curve (green line).x(t)tWhen the velocity is constant, the average velocity over any time interval is equal to the instantaneous velocity at any time.Instantaneous speed: Magnitude of the instantaneous velocity. Example: car speedometer.- Scalar quantityAverage velocity (or average acceleration) always refers to an specific time interval. Instantaneous velocity (acceleration) refers to an specific instant of time.Slope of the particle’s position-time curve at a given instant of time. V is tangent to x(t) when ∆t0Instantaneous velocity:TimePosition5III. AccelerationVAverage acceleration: Ratio of a change in velocity ∆v to the time interval∆t in which the change occurs.- Vector quantity- Dimensions [L]/[T]2, Units: m/s2- The average acceleration in a “v-t” plot is the slopeof a straight line connecting points corresponding totwo different times.)5.2(1212tvttvvaavg∆∆=−−=ttt)6.2(lim220dtxddtdvtvat==∆∆=→∆Instantaneous acceleration: Limit of the average acceleration as ∆t approaches zero.- Vector quantity- The instantaneous acceleration is the slope of the tangent line (v-t plot) at a particular time. (green line in B)- Average acceleration: blue line.- When an object’s velocity and acceleration are in the same direction (same sign), the object is speeding up.- When an object’s velocity and acceleration are in the opposite direction, the object is slowing down.6Example (2): x(t)=At2  v(t)=2At  a(t)=2A ; At t=0s, v(0)=0 but a(0)=2AExample (3):Example (1): v1= -25m/s ; v2= 0m/s in 5s  particle slows down, aavg= 5m/s2- An object can have simultaneously v=0 and a≠0- Positive acceleration does not necessarily imply speeding up, and negative acceleration slowing down.- The car is moving with constant positive velocity (red arrows maintaining same size)  Acceleration equals zero.Example (4):- Velocity and acceleration are in the same direction, “a” is uniform (blue arrows of same length)  Velocity is increasing (red arrows are getting longer).+ acceleration+ velocityExample (5):- acceleration+ velocity- Acceleration and velocity are in opposite directions.- Acceleration is uniform (blue arrows same length).- Velocity is decreasing (red arrows are getting shorter).700−−==tvvaaavg- Equations for motion with constant acceleration:)11.2()(2)2(2)(2)10.2(),7.2()10.2(2)9.2(),8.2()9.2(2)7.2(2)8.2()7.2(020220222002220220000000xxavvatxxatavtvatavvattvxxatvvandvvvtvxxtxxvatvvavgavgavgavg−+=→−−++=++=→+=−→+=→+=+=→−=+=ttt missingIV. Motion in one dimension with constant acceleration- Average acceleration and instantaneous acceleration are equal.00−−==tvvaaavgtPROBLEMS - Chapter 2P1. A red car and a green car move toward each other in adjacent lanes and parallel to The x-axis. At time t=0, the red car is at x=0 and the green car at x=220 m. If the red car has a constant velocity of 20km/h, the cars pass each other at x=44.5 m, and if it has a constant velocity of 40 km/h, they pass each other at x=76.6m. What are (a) the initial velocity, and (b) the acceleration of the green car?smkmmshhkm/11.1111036001403=⋅⋅xd=220 mXr1=44.5 mXr2=76.6mOvr1=20km/hvr2=40km/hXg=220m)2(221)1(000attgvgxgxtrvrxrx++=+=ssmmttrvrxssmmttrvrx9.6/11.116.768/55.55.4422221111==→===→=gggggggassvtatvxrxassvtatvxrxggg22)8(5.0)8(2205.44)9.6(5.0)9.6(2206.760215.01010225.0202⋅−⋅−=−→−=−⋅−⋅−=−→−=−⋅−⋅−ag= 2.1 m/s2 v0g = 13.55 m/scThe car moves to the left (-) in my reference system  a<0, v<08P2: At the instant the traffic light turns green, an automobile starts with a constant acceleration a of 2.2 m/s2. At the same instant, a truck, traveling with constant speed