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    Chapter 2: Motion in One Dimension: 2.1 Position, Velocity, and Speed

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    Faris Alasmari
    This chapter discusses one-dimensional motion including: 1) Position, displacement, distance, velocity, speed, and acceleration. Velocity and speed are different concepts. 2) Instantaneous velocity and acceleration which are the derivatives of position and velocity with respect to time. 3) Kinematic equations that relate the variables of motion for constant acceleration situations. 4) Freely falling objects which experience a constant downward acceleration due to gravity near Earth's surface. Several example problems are worked through applying the concepts.

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    Chapter 2: Motion in One Dimension: 2.1 Position, Velocity, and Speed

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    Faris Alasmari
    102 views12 pages
    This chapter discusses one-dimensional motion including: 1) Position, displacement, distance, velocity, speed, and acceleration. Velocity and speed are different concepts. 2) Instantaneous velocity and acceleration which are the derivatives of position and velocity with respect to time. 3) Kinematic equations that relate the variables of motion for constant acceleration situations. 4) Freely falling objects which experience a constant downward acceleration due to gravity near Earth's surface. Several example problems are worked through applying the concepts.

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    Motion in One Dimension

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    This chapter discusses one-dimensional motion including: 1) Position, displacement, distance, velocity, speed, and acceleration. Velocity and speed are different concepts. 2) Instantaneous velocity and acceleration which are the derivatives of position and velocity with respect to time. 3) Kinematic equations that relate the variables of motion for constant acceleration situations. 4) Freely falling objects which experience a constant downward acceleration due to gravity near Earth's surface. Several example problems are worked through applying the concepts.

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    102 views12 pages

    Chapter 2: Motion in One Dimension: 2.1 Position, Velocity, and Speed

    Uploaded by

    Faris Alasmari
    This chapter discusses one-dimensional motion including: 1) Position, displacement, distance, velocity, speed, and acceleration. Velocity and speed are different concepts. 2) Instantaneous velocity and acceleration which are the derivatives of position and velocity with respect to time. 3) Kinematic equations that relate the variables of motion for constant acceleration situations. 4) Freely falling objects which experience a constant downward acceleration due to gravity near Earth's surface. Several example problems are worked through applying the concepts.

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    Chapter 2: Motion in One Dimension

    2.1 Position, Velocity, and Speed

    The particle model: we describe the moving object as a particle regardless


    of its size

    A particle’s position is the location of the particle with respect to a chosen


    reference point that we can consider to be the origin of a coordinate system.
    position–time graph

    The displacement of a particle is defined as its change in position in some


    time interval.

    Distance is the length of a path followed by a particle

    In physics the terms speed and velocity are not the same

    The average velocity = ∆x / ∆t

    The average speed = total distance / total time

    Displacement and velocity are vector quantities (require the specification of


    both direction and magnitude).
    By contrast, a scalar quantity has a numerical value and no direction.
    Example 2.1

    Find the displacement, average velocity, and average speed of the car in the
    previous figure between positions A and F. (assume that the details of the
    car’s position are described by the previous curve)

    Average velocity= -1.7 m/s

    Average speed = 2.5 m/s

    2.2 Instantaneous Velocity and Speed

    We need to know the velocity of a particle at a particular instant in time,


    rather than the average velocity over a finite time interval.

    The instantaneous velocity ʋx equals the limiting value of the ratio ∆x/∆t as
    ∆t approaches zero

    the derivative of x with respect to t


    The instantaneous speed of a particle is defined as the magnitude of its
    instantaneous velocity

    From here on, velocity means instantaneous velocity and speed means
    instantaneous speed.
    Example 2.2

    Consider the following one-dimensional motions: (A) A ball thrown directly


    upward rises to a highest point and falls back into the thrower’s hand. (B) A
    race car starts from rest and speeds up to 100 m/s. (C) A spacecraft drifts
    through space at constant velocity. Are there any points in the motion of
    these objects at which the instantaneous velocity has the same value as the
    average velocity over the entire motion?
    If so, identify the point(s).

    See page 30

    Example 2.3 (see page 30)

    A particle moves along the x axis. Its position varies with time according to
    the expression x = -4t + 2t2 where x is in meters and t is in seconds. The
    position–time graph for this motion is shown in the following figure.

    (A) Determine the displacement of the particle in the time intervals t = 0


    to t = 1 s and t = 1 s to t = 3 s.
    (B) Calculate the average velocity during these two time intervals.
    (C) Find the instantaneous velocity of the particle at t = 2.5 s

    (Note that the particle moves in the negative x direction for the first second
    of motion, is momentarily at rest at the moment t = 1 s, and moves in the
    positive x direction at times t> 1 s)
    Example (from a previous exam)

    The motion of two cars A and B moving in a straight line is described by:

    xA(t)= 4t2 and xB(t)= 6-5t+5t2

    the two cars will reach the same velocity at a time (in seconds):

    (a) 5.4 s (b) 1 s (c) 1.3 s (d) 0.28 s (e) 2.5 s


    2.3 Acceleration

    The average acceleration

    The instantaneous acceleration

    The acceleration can be positive or negative (vector)


    Negative acceleration does not necessarily mean that an object is slowing
    down.

    When the object’s velocity and acceleration are in the same direction, the
    object is speeding up. On the other hand, when the object’s velocity and
    acceleration are in opposite directions, the object is slowing down.
    Example 2.5 see page 33

    The velocity of a particle moving along the x axis varies in time according to
    the expression v=(40-5t2) m/s, where t is in seconds.
    (a) Find the average acceleration in the time interval t =0 to t = 2.0 s.
    (-10 m/s2)
    (b) Determine the acceleration at t = 2.0 s.
    -20 m/s2

    2.5 One-Dimensional Motion with Constant Acceleration

    The kinematic equations for motion of a particle under constant acceleration:

    Try to prove these equations (in lecture)


    Example 2.7
    A jet lands on an aircraft carrier at 140 mi/h (~63 m/s).

    (A)What is its acceleration (assumed constant) if it stops in 2.0 s.


    (-31m/s2)
    (B) If the plane touches down at position xi=0, what is the final position
    of the plane?
    63m

    Example 2.8 see page 39


    A car traveling at a constant speed of 45.0 m/s passes a trooper hidden
    behind a billboard. One second after the speeding car passes the billboard,
    the trooper sets out from the billboard to catch it, accelerating at a constant
    rate of 3.00 m/s2. How long does it take her to overtake the car?
    ~31s
    2.6 Freely Falling Objects

    A freely falling object is any object moving freely under the influence of
    gravity alone, regardless of its initial motion. (upward or downward)

    Any freely falling object experiences an acceleration directed downward,


    regardless of its initial motion.

    In the absence of air resistance, all objects dropped near the Earth’s surface
    fall toward the Earth with the same constant acceleration (g) under the
    influence of the Earth’s gravity

    At the Earth’s surface, the value of g is approximately 9.80 m/s2 (decreases


    with increasing altitude)

    for freely falling objects, the motion is in the vertical direction and the
    acceleration is downward
    ay = g = -9.80 m/s2.

    Example 2.10 (p 41)

    A ball is tossed straight up at 25 m/s. Estimate its velocity at 1 second


    intervals.
    Example

    Which values represent the ball’s vertical velocity and acceleration at points
    A, C and E in the following figure?

    (a) vy= 0, ay = - 9.80 m/s2


    (b) vy= 0, ay = 9.80 m/s2
    (c) vy= 0, ay= 0
    (d) vy = - 9.80 m/s, ay = 0

    Example 2.12 (p 43)


    A stone thrown from the top of a building is given an initial velocity of 20
    m/s straight upward. The building is 50 m high, and the stone just misses the
    edge of the roof on its way down. Using tA = 0 as the time the stone leaves
    the thrower’s hand at position A, determine:

    (A) The time at which the stone reaches its maximum height
    (B) The maximum height,
    (C) The time at which the stone returns to the height from which it was
    thrown,
    (D) The velocity of the stone at this instant
    (E) The velocity and position of the stone at t = 5 s.
    For illustration, see the following figure

    Try to solve these problems (4-6-16-21-25-29-33-40-42-48-52) from chapter


    two (p50-54) (some of them will be solved in lectures)

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