1D - 1 Motion 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 velocity distance 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 +/– sign + = going right → always! – = going left ← Objects 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 t∆≡=∆ fi 21fi 21xx xx xvtt tt t−−===−−∆∆ ∆x = xfinal – xinitial = displacement (can be + or – ) B vB = +10 m/s vA = –10 m/s A x 0 x1 x2 x 0 (initial) (final) 1/5/2010 Dubson Notes © University of Colorado at Boulder1D - 2 Notice that ∆ (delta) always means "final minus initial". xvt∆=∆ is the slope of a graph of x vs. t Review: Slope of a line Suppose we travel along the x-axis, in the positive direction, at constant velocity v: x 0 startx ∆x ∆t x2t x1t1t2∆y∆xslope = rise run = = ∆x∆t = v y-axis is x, x-axis is t . y x ∆y ∆x ∆y∆xslope = rise run = (x2, y2) y2 – y1 = x2 – x1 (x1, y1) x y (+) slope y y 0 slope x (–) slope x 1/5/2010 Dubson Notes © University of Colorado at Boulder1D - 3 Now, let us travel in the negative direction, to the left, at constant velocity. start Note that v = constant ⇒ slope of x vs. t = constant ⇒ graph of x vs. t is a straight line But 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 t0xdvlimtd∆→xt≡≡∆ = 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) is defined as the slope of the tangent line: ∆t0dx xlimdt t∆→∆≡∆. x ∆x t∆t ∆x ∆t x 0 x ∆xslope = v = < 0 ∆t t ∆t ∆x < 0 x slower slope > 0 (fast) tslope = 0 (stopped) 1/5/2010 Dubson Notes © University of Colorado at Boulder1D - 4 Acceleration If the velocity is changing, then there is non-zero acceleration. efinition: acceleration = time rate of change of velocity = derivative of velocity with respect to me Dti In 1D: instantaneous acceleration t0alimtdt∆→vdv∆≡=∆ average acceleration over a non-infinitesimal time interval ∆t : vat∆≡∆ units of a = 2[a]ss== m/s mometimes I will be a bit sloppy and just write Svat∆=∆, 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. x t ∆x ∆t tangent line x t v = dx/dt t slowfast 1/5/2010 Dubson Notes © University of Colorado at Boulder1D - 5 fi 21vv vva−−∆=== • v increasing (becoming more positive) ⇒ a > 0 • v decreasing (becoming more negative) ⇒ a < 0 n 1D, c vs. t (just like v = slope of x vs. t ) xamp vdvfi 21dt t t t t t∆− − • v = constant ⇒ ∆v = 0 ⇒ a = 0 I ac eleration a is the slope of the graph of v E les of constant acceleration in 1D on next page... 1/5/2010 Dubson Notes © University of Colorado at Boulder1D - 6 Examples of constant acceleration in 1D 1 Situation IAn object starts at rest, then moves to the right (+ direction) with constant acceleration, going faster and faster. 2 4 3 v ∆v ∆t 4 a > 0, a = constant 3 (a constant, since v vs. t is straight ) 2 t 1 1 Situation IIAn object starts at rest, then moves to the left (– direction) with constant acceleration, going faster and faster. 4 3 2 v ∆v ∆t t 1 a < 0, a = constant 2 ( since v vs. t has constant, negative slope ) 3 4 Situation III3 v t a < 0, a = constant !! ( since v vs. t has constant, negative slope )4 5 1 2 3 5 1 2 4 1/5/2010 Dubson Notes © University of Colorado at Boulder1D - 7 The direction of the acceleration For 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 acceleration to the left. But what is it, exactly, that is pointing right or left when we talk about the direction of the acceleration? Acceleration and velocity are both examples of vector quantities. They are mathematical objects that have both a magnitude (size) and a direction. We often represent vector quantities by putting a little arrow over the symbol, like v or aKK. direction of ≠ direction of aKvK direction of = the direction toward which the velocity is aKtending ≠ direction of vK Reconsider Situation I (previous page) ( This has been a preview of Chapter 3, dvadt=KK ) Our mantra: " Acceleration is not velocity, velocity is not acceleration." Situation II: v2 ∆vv1 In both situations I and II, ∆v is to the left, so acceleration a is to the left Situation III: v2 ∆vv1 1 2 1 is an earlier time, 2 is a later time v1 = velocity at time 1 = vinitv2 = velocity at time 2 = vfinal∆v = "change vector" = how v1 must be "stretched" to change it into v2 ∆vv1 direction of a = direction of ∆v v2 1/5/2010 Dubson Notes © University of Colorado at Boulder1D - 8 Constant 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) (v, t) ovv a=+to) (b) (x, t) 2ooxx vt(1/2)at=+ + (c) (v, x) 22ovv2a(xx=+ − (d) ovvv2+= xo , vo = initial position, initial velocity x, v = position, velocity at time t Reminder: all of