1D - 1Motion in one dimension (1D) [ Chapter 2 in Wolfson ]In this chapter, we study speed, velocity, and acceleration for motion in one-dimension. One dimensional motion is motion along a straight line, like the motion of a glider on an airtrack.speed and velocitydistance traveled dspeed , s = , units are m/s or mph or km/hr or...time elapsed t=speed s and distance d are both always positive quantities, by definition.velocity = speed + direction of motion Things that have both a magnitude and a direction are called vectors. More on vectors in Ch.3.For 1D motion (motion along a straight line, like on an air track), we can represent the direction of motion with a +/– signObjects A and B have the same speed s = |v| = +10 m/s, but they have different velocities.If the velocity of an object varies over time, then we must distinguish between the average velocity during a time interval and the instantaneous velocity at a particular time.Definition: change in position xaverage velocity = v change in time tD� =Df i 2 1f i 2 1x x x x xvt t t t t- - D= = =- - Dx = xfinal – xinitial = displacement (can be + or – )1/14/2019 Dubson Notes University of Colorado at Boulder+ = going right – = going left always!BvB = +10 m/svA = –10 m/sAx0(final)(initial)x0x1x21D - 2Notice that (delta) always means "final minus initial".xvtD=D is the slope of a graph of x vs. tReview: Slope of a lineSuppose we travel along the x-axis, in the positive direction, at constant velocity v:1/14/2019 Dubson Notes University of Colorado at Boulderxyyxyxslope = riserun= xy(+) slopexy(–) slopexy0 slopey2 – y1= x2 – x1(x1, y1)(x2, y2)x0startxxtx2tx1t1t2yxslope = riserun= = xt= vy-axis is x, x-axis is t .1D - 3Now, let us travel in the negative direction, to the left, at constant velocity.Note that v = constant slope of x vs. t = constant graph of x vs. t is a straight lineBut what if v constant? If an object starts out going fast, but then slows down and stops...The slope at a point on the x vs. t curve is the instantaneous velocity at that point.Definition: instantaneous velocity = velocity averaged over a very, very short (infinitesimal) time interval t 0x d xv limt d tD �D� �D = slope of tangent line. In Calculus class, we would say that the velocity is the derivative of the position with respect to time. The derivative of a function x(t) isdefined as the slope of the tangent line: t 0d x xlimd t tD �D�D.1/14/2019 Dubson Notes University of Colorado at Boulderx0startxx < 0ttslope = v = xt< 0xslowerslope > 0 (fast)tslope = 0 (stopped)xxttxt1D - 4AccelerationIf the velocity is changing, then there is non-zero acceleration.Definition: acceleration = time rate of change of velocity = derivative of velocity with respect totimeIn 1D: instantaneous acceleration t 0v d va limt d tD �D� =D average acceleration over a non-infinitesimal time interval t : vatD�Dunits of a = 2m / s m[a]s s= = 1/14/2019 Dubson Notes University of Colorado at Boulderxtxttangent linextv = dx/dttslowfast1D - 5Sometimes I will be a bit sloppy and just write vatD=D, where it understood that t is either a infinitesimal time interval in the case of instantaneous a or t is a large time interval in the case of average a. f i 2 1f i 2 1v v v vd v vad t t t t t t- -D= = =D - -; v = constant v = 0 a = 0 v increasing (becoming more positive) a > 0 v decreasing (becoming more negative) a < 0In 1D, acceleration a is the slope of the graph of v vs. t (just like v = slope of x vs. t )Examples of constant acceleration in 1D on next page... 1/14/2019 Dubson Notes University of Colorado at Boulder1D - 6Examples of constant acceleration in 1D 1/14/2019 Dubson Notes University of Colorado at Boulder1Situation Ivvtta > 0, a = constant(a constant, since v vs. t is straight )An object starts at rest, then moves to the right (+ direction) with constant acceleration, going faster and faster.23412341Situation IIvvtt a < 0, a = constant( since v vs. t has constant, negative slope )An object starts at rest, then moves to the left (– direction) with constant acceleration, going faster and faster.23412343Situation IIIvt a < 0, a = constant !!( since v vs. t has constant, negative slope )4512351241D - 7The direction of the accelerationFor 1D motion, the acceleration, like the velocity, has a sign ( + or – ). Just as with velocity, we say that positive acceleration is acceleration to the right, and negative acceleration is accelerationto the left. But what is it, exactly, that is pointing right or left when we talk about the direction ofthe acceleration? Acceleration and velocity are both examples of vector quantities. They are mathematical objectsthat have both a magnitude (size) and a direction. We often represent vector quantities by putting a little arrow over the symbol, like v or avv.direction of av direction of vv direction of av = the direction toward which the velocity is tending direction of vvReconsider Situation I (previous page) ( This has been a preview of Chapter 3, d vad t=vv )Our mantra: " Acceleration is not velocity, velocity is not acceleration."1/14/2019 Dubson Notes University of Colorado at Boulder121 is an earlier time, 2 is a later timev1 = velocity at time 1 = vinitv2 = velocity at time 2 = vfinalv = "change vector" = how v1 must be "stretched" to change it into v2 v1 v2 vdirection of a = direction of vSituation II:v2 vv1 In both situations I and II, v is to the left, so acceleration a is to the leftSituation III:v2 vv11D - 8Constant acceleration formulas (1D)In the special case of constant acceleration (a = constant), there are a set of formulas that relate position x, velocity v, and time t to acceleration a.formula relates(a) ov v a t= +(v, t)(b)2o ox x v t (1/ 2) a t= + +(x, t)(c)2 2o ov v 2a (x x )= + -(v, x)(d)ov vv2+=xo , vo = initial position, initial velocity x, v = position, velocity at time tReminder: all of these formulas are only valid if a = constant, so these are special case formulas. They are not laws. (Laws are always true.)Proof of formula (a) ov v a t= +. Start with definition d vad t= . In the case of constant acceleration, 2 12 1v vva at t t-D= = =D -Since a = constant,
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