Would you risk your live driving drunk? Intro

Transcription

1 Martha Casquete

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3 Would you risk your lie driing drunk? Intro

4 Motion Position and displacement Aerage elocity and aerage speed Instantaneous elocity and speed Acceleration Constant acceleration: A special case Free fall acceleration

5 Assignments: For next class: Finish reading Ch., read Chapter 3 (Vectors) HW3 Set due next Tuesday, 9/10 Pg. 48 5: 5, 11, 18, 40, 49 (8 th edition) Question/Obseration Thursdays Research Q/O Tuesdays with HW (due date Tuesdays)

6 Units: Length, Mass, and Time Order of magnitude calculations Dimensional Analysis Conersion of Units Significant digits (on your own)

7 Need to know something from your experience: Aerage length of french fry: 3 inches or 8 cm, 0.08 m Earth to moon distance: 50,000 miles In meters: 1.6 x.5 X 10 5 km = 4 X 10 8 m ff m m

8 Example What is the dimension of acceleration? a = / t ( elocity / time ) a = ( L / T ) / T = L / T

9 Eerything moes! Motion is one of the main topics of this course Simplification: Moing object is a particle or moes like a particle: point object Simplest case: Motion along straight line, 1 dimension Continental - McAllen La Guardia -NY

10 What is motion? Change of position oer time. How can we represent position along a straight line? Position definition: Defines a starting point: origin (x = 0), x relatie to origin Direction: positie (right or up), negatie (left or down) It depends on time: t = 0 (start clock), x(t=0) does not hae to be zero. Position has units of [Length]: meters. x = +.5 m x = - 3 m

11 A ector quantity is characterized by haing both a magnitude and a direction. Displacement, Velocity, Acceleration, Force Denoted in boldface type with an arrow oer the top., a, F... A scalar quantity has magnitude, but no direction. Distance, Mass, Temperature, Time For the motion along a straight line, the direction is represented simply by + and signs. + sign: Right or Up. - sign: Left or Down.

12 Any motion inoles three concepts Displacement Velocity Acceleration

13 Displacement is a change of position in time. Displacement: It is a ector quantity. x ( t f ) xi ( ti) It has both magnitude and direction: + or - sign It has units of [length]: meters. x 1 (t 1 ) = +.5 m x (t ) = -.0 m Δx = -.0 m -.5 m = -4.5 m x f x 1 (t 1 ) = m x (t ) = m Δx = +1.0 m m = +4.0 m

14 Position is usually measured and referenced to an origin: 10 meters At time= 0 seconds Joe is 10 meters to the right of the lamp origin = lamp positie direction = to the right of the lamp position ector : 10 meters -x +x O Joe

15 One second later Joe is 15 meters to the right of the lamp Displacement is just change in position. x = x f - x i 10 meters 15 meters O x i x f Joe x = x f - x i = 5 meters t = t f - t i = 1 second

16 Displacement in space From A to B: Δx = x B x A = 5 m 30 m = m From A to C: Δx = x c x A = 38 m 30 m = 8 m Sometimes displacement is confused with distance. It is important that you understand the difference between distance and displacement.

17 Distance is the length of a path followed by a particle from A to B: d = x B x A = 5 m 30 m = m from A to C: d = x B x A + x C x B = m + 38 m 5 m = 36 m Displacement is not Distance.

18 Eery day, I trael a distance of 108 miles from Brownsille/Rancho Viejo to Edinburg and back. Eery day, at 8:0pm I noticed that I hae displaced myself zero miles. Where am I? What is the distance between one side of the wall and the other in the classroom? If I pace it, what will be my displacement? Displacement is not Distance.

19 Velocity is the rate of change of position. Velocity is a ector quantity. Velocity has both magnitude and direction. Velocity has a unit of [length/time]: meter/second. Definition: x x Aerage elocity x f i ag t t Aerage speed Instantaneous elocity s ag total distance t lim t 0 x t dx dt

20 Aerage elocity ag x t x f x t i It is slope of line segment. Dimension: [length/time]. SI unit: m/s. It is a ector. Displacement sets its sign.

21 Aerage speed s ag total distance t Dimension: [length/time], m/s. Scalar: No direction inoled. Not necessarily close to V ag : S ag = (6m + 6m)/(3s+3s) = m/s V ag = (0 m)/(3s+3s) = 0 m/s

22 Under which of the following conditions is the magnitude of the aerage elocity of a particle moing in one dimension smaller than the aerage speed oer some time interal? a) a particle moes in the +x direction without reersing. b) a particle moes in the x direction without reersing c) a particle moes in the +x direction and then reerses the direction of its motion. d) there are no conditions for which this is true

23 Instantaneous means at some gien instant. The instantaneous elocity indicates what is happening at eery point of time. Limiting process: Chords approach the tangent as Δt => 0 Slope measure rate of change of position Instantaneous elocity: x dx lim It is a ector quantity. t 0 t dt Dimension: [Length/time], m/s. It is the slope of the tangent line to x(t). Instantaneous elocity (t) is a function of time.

24 A cheetah is crouched in ambush 0 m to the east of an obserer s blind. At time t= 0 the cheetah charges an antelope in a clearing 50 m east of the obserer. The cheetah runs along a straight line. Later analysis of a ideotape shows that during the first.0 s of the attack, the cheetah s coordinator x aries with time according to the equation x = 0m + (5.0m/s )t a) Find the displacement of the cheetah during the interal between t1 = 1.0 s and t =.0 s b) Find the aerage elocity during the same time interal. c) Find the aerage elocity at time t1 = 1.0 s by taking t = 0.1 s. d) Derie a general expression for the instantaneous elocity as a function of time, and from it find at t = 1.0 s and =.0 s.

25 Uniform elocity is constant elocity The instantaneous elocities are always the same, all the instantaneous elocities will also equal the aerage elocity Begin with then x x f xi x f x x f x i x(t) x t x x t t (t) x i 0 t 0 t t i t f

26 Changing elocity (non-uniform) means an acceleration is present. Acceleration is the rate of change of elocity. Acceleration is a ector quantity. Acceleration has both magnitude and direction. Acceleration has a unit of [length/time ]: m/s. Definition: Aerage acceleration Instantaneous acceleration a ag t t f f t i i a lim t0 t d dt d dt dx dt d dt

27 Aerage acceleration a ag t t f f t i i Velocity as a function of time f ( t) i a ag t When the sign of the elocity and the acceleration are the same (either positie or negatie), then the speed is increasing When the sign of the elocity and the acceleration are in the opposite directions, the speed is decreasing Aerage acceleration is the slope of the line connecting the initial and final elocities on a elocity-time graph

28 The limit of the aerage acceleration as the time interal goes to zero a lim t0 t d dt d dt dx dt d dt When the instantaneous accelerations are always the same, the acceleration will be uniform. The instantaneous acceleration will be equal to the aerage acceleration September 8, 008

29 Velocity and acceleration are in the same direction f ( t) i Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Positie elocity and positie acceleration at

30 Position is a function of time: Velocity is the rate of change of position. Acceleration is the rate of change of elocity. lim t 0 d dt x t dx dt t x x(t) Position Velocity Acceleration Graphical relationship between x,, and a a lim t0 d dt d dt

31 0 at Uniform acceleration is constant Kinematic Equations x t 1 ( 0 ) t x 0t at 1 0 ax

32 Gien initial conditions: a(t) = constant = a, (t=0) = 0, x(t=0) = x 0 Start with 0 at We hae a ag t t t t 0 t Shows elocity as a function of acceleration and time Use when you don t know and aren t asked to find the displacement 0 a

33 Gien initial conditions: a(t) = constant = a, (t=0) = 0, x(t=0) = x 0 Start with ag x x t 0 x t Since elocity change at a constant rate, we hae x t 1 ag ( 0 ) t Gies displacement as a function of elocity and time Use when you don t know and aren t asked for the acceleration

34 Gien initial conditions: a(t) = constant = a, (t=0) = 0, x(t=0) = x 0 Start with We hae x x x 0 x t 1 ag ( 0 ) t 0 1 at Gies displacement as a function of time, initial elocity and acceleration Use when you don t know and aren t asked to find the final elocity 0 at 1 1 x ( 0 ) t ( 0 0 at) t

35 Gien initial conditions: a(t) = constant = a, (t=0) = 0, x(t=0) = x 0 Start with We hae Gies elocity as a function of acceleration and displacement Use when you don t know and aren t asked for the time ) ( x x a x a a t 0 a t t a ag 0 a a t x ) )( ( 1 ) (

36 Read the problem Draw a diagram Choose a coordinate system, label initial and final points, indicate a positie direction for elocities and accelerations Label all quantities, be sure all the units are consistent Conert if necessary Choose the appropriate kinematic equation Sole for the unknowns You may hae to sole two equations for two unknowns Check your results Estimate and compare Check units

37 An electron in a cathode-ray tube accelerates from a speed of.00 X 10 m/s oer 1.50 cm. (a) in what time interal does the electron trael this 1.50 cm? (b) What is the acceleration

38 y Earth graity proides a constant acceleration. Most important case of constant acceleration. Free-fall acceleration is independent of mass. Magnitude: a = g = 9.8 m/s Direction: always downward, so a g is negatie if define up as positie, a = -g = -9.8 m/s Try to pick origin so that x i = 0 Galileo Galilei (

39 x 0 Two important equation: x 0 x 0 gt 0 t 1 gt Begin with t 0 = 0, 0 = 0, x 0 = 0 So, t = x /g same for two balls! Assuming the leaning tower of Pisa is 150 ft high, neglecting air resistance, t = ( /9.8) 1/ = 3.05 s

40 What do you need? Ruler Pencil Paper Brain Volunteer with two fingers

41 This is the simplest type of motion It lays the groundwork for more complex motion Kinematic ariables in one dimension Position x(t) m L Velocity (t) m/s L/T Acceleration a(t) m/s L/T All depend on time All are ectors: magnitude and direction ector: Equations for motion with constant acceleration: missing quantities 0 at x x 0 1 x x0 0t at 0 a( x x0) t 1 x x0 ( 0) t a 1 x x 0 t at 0 September 8, 008