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    Dynamics MDB 2043

    Rectilinear Kinematics: Erratic Motion

    Guided Learning Activity

    May 2016 Semester

    Lesson Outcomes

    At the end of this lecture you should be able to:

    Determine position, velocity, and acceleration of a

    particle using graphs.

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    Example #1 (continued)

    Similarly, the a-t graph can be constructed by finding the slope at various points

    along the v-t graph.

    when 0 < t < 5 s; a0-5 = dv/dt = d(6t)/dt = 6 m/s2

    when 5 < t < 10 s; a5-10 = dv/dt = d(30)/dt = 0 m/s2

    a-t graph

    a(m/s2)

    t(s)

    6

    5 10

    Example #2

    Find slopes of the v-t curve and draw the a-t graph.

    Find the area under the curve. It is the distance traveled.

    Finally, calculate average speed (using basic definitions!).

    Given: The v-t graph shown.

    Find: The a-t graph, average

    speed, and distance

    traveled for the 0 - 90 s

    interval.

    Plan:

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    Example #2 (continued)

    Solution:

    Find the at graph:

    For 0 t 30 a = dv/dt = 1.0 m/s

    For 30 t 90 a = dv/dt = -0.5 m/s

    a-t graph

    -0.5

    1

    a(m/s)

    30 90 t(s)

    Example #2 (continued)

    Now find the distance traveled:

    Ds0-30 = v dt = (1/2) (30)2 = 450 m

    Ds30-90 = v dt= (1/2) (-0.5)(90)2 + 45(90) (1/2) (-0.5)(30)2 45(30)

    = 900 m

    s0-90 = 450 + 900 = 1350 m

    vavg(0-90) = total distance / time

    = 1350 / 90

    = 15 m/s

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    Example #3

    A motorcycle starts from rest and travels on a straight road with a constant

    acceleration of 5 m/s2 for 8 sec, after which it maintains a constant speed for 2

    sec. Finally it decelerates at 7 m/s2 until it stops. Plot a-t, v-t diagrams for theentire motion.Determine the total distance travelled.

    Sketch a-t diagram from the known accelerations, thus

    5

    -7

    )'10(7

    )108(0

    )80(5

    tt

    st

    st

    a

    (segment I)

    (segment II)

    (segment III)

    Since dv=adt, the v-t diagram is determined by integratingthe straight line segments of a-t diagram. Using the initial

    condition t=0, v=0 for segment I, we have

    st 80

    tv

    dtdv005 tv 5

    When t =8 s, v =58= 40m/s. Using this as the initial condition

    for segment II, thus

    st 108

    tv

    dtdv8400 smv /40

    Similarly, for segment III

    '10 tt

    tv

    dtdv1040

    )7( 1107tv

    a(m/s2)

    t (s)

    8 10 t' (=15.71)

    a-t Diagram

    How can you determine t?

    When v=0 (i.e. motorcycle stops)

    110'70 t

    )71.1510(1107

    )108(40

    )80(5

    stt

    st

    stt

    v

    st 71.15'

    Thus, the velocity as the function of time can

    be expressed as

    The total distance travelled (using the area under v-t diagram)

    mssss 2.3544071.52

    1

    4024082

    1321

    v (m/s)

    t (s)

    40

    8 10 15.71

    v-t Diagram

    s1 s2 s3

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    Example #4

    A test projectile is fired horizontally into a viscous liquid with a velocity

    v0.The retarding force is proportional to the square of the velocity, so

    that the acceleration becomes a=-kv2. Derive expressions for distanceD travelling in the liquid and the corresponding time t required to reduce

    the velocity to v0/2.Neglect any vertical motion.

    Note the acceleration a is non-constant.

    Using dskvadsvdv 2

    22

    20

    0

    0

    0

    0

    v

    v

    v

    v

    D

    kv

    dv

    kv

    vdvds

    kkv

    v

    kk

    vD

    v

    v

    693.02ln2ln1ln

    0

    0

    2

    0

    0

    Using2

    kv

    dt

    dva

    tv

    v

    dtkv

    dv

    0

    22

    0

    0

    0

    2 1110

    0kvvk

    t

    v

    v

    Example #5

    The acceleration of a particle which moves in thepositive s-direction varies with its position as

    shown. If the velocity of the particle is 0.8 m/s

    when s=0, determine the velocities v of the particle

    when s=0.6 and 1.4 m.

    ax (m/s2

    )

    s (m)

    0.4

    0.2

    0.4 0.8 1.2

    Using

    22

    2

    0

    22

    0

    0

    0

    vvvvdvads

    v

    v

    v

    v

    s

    smadsvv /17.102.04.04.0)4.02.0(2

    1)4.04.0(28.02 2

    4.1

    0

    2

    0

    For x=1.4m

    Where v0=0.8 m/sArea under ax-x curve

    (0x 1.4)

    For x=0.6m

    1.40.6

    smadsvv /05.12.0)4.03.0(2

    1)4.04.0(28.02 2

    6.0

    0

    2

    0

    Area under ax-x curve(0x 0.6)

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    Example #6

    The v-s diagram for a testing vehicle travelling on a

    straight road is shown. Determine the acceleration

    of the vehicle at s=50 m and s=150 m. Draw thea-s diagram.

    v (m/s)

    s (m)

    100 200

    8

    Since the equations for segments of v-s diagram are given,

    we can use ads=vdv to determine a-s diagram.

    ms 1000

    ssds

    ds

    ds

    dvva 0064.0)08.0()08.0(

    ms 200100

    28.10064.0)1608.0()1608.0( ssds

    dsa

    1608.0 sv

    sv 08.0

    When s=50 m, then (acceleration in segmentI)2

    /32.0500064.0 sma

    When s=150 m, then (deceleration in segmentII)2

    /32.028.11500064.0 sma

    a(m/s2)

    100 200 s (m)

    0.64

    -0.64

    1. The slope of a v-t graph at any instant represents instantaneous

    A) velocity. B) acceleration.

    C) position. D) jerk.

    2. Displacement of a particle in a given time interval equals the

    area under the ___ graph during that time.

    A) a-t B) a-s

    C) v-t D) s-t

    Summary Questions

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    3. If a particle starts from rest and

    accelerates according to the graph

    shown, the particles velocity at

    t = 20 s is

    A) 200 m/s B) 100 m/s

    C) 0 D) 20 m/s

    4. The particle in Problem 3 stops moving at t = _______.

    A) 10 s B) 20 s

    C) 30 s D) 40 s

    Summary Questions (continued)

    5. If a car has the velocity curve shown, determine the time t

    necessary for the car to travel 100 meters.

    A) 8 s B) 4 s

    C) 10 s D) 6 s

    t

    v

    6 s

    75

    t

    v

    6. Select the correct a-t graph for the velocity curve shown.

    A) B)

    C) D)

    a

    t

    a

    t

    a

    t

    a

    t

    Summary Questions (continued)

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    References:

    R.C. Hibbeler, Engineering Mechanics: Dynamics,

    SI 13th Edition, Prentice-Hall, 2012.