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    Lesson 2: Motion in One Dimension: Understanding How Things Move

    pj316
    4–6 minutes
    ,

    Introduction

    Motion is one of the most fundamental concepts in physics.

    A car traveling down a highway, blood flowing through an artery, a package moving through a warehouse, Earth orbiting the Sun, and electrons moving inside an atom are all examples of motion.

    Before studying forces, energy, quantum mechanics, or relativity, we must first learn how to describe motion mathematically.

    In this lesson we will study:

    • Position
    • Distance
    • Displacement
    • Speed
    • Velocity
    • Acceleration
    • Motion graphs
    • Constant acceleration
    • Free fall

    What Is Motion?

    An object is in motion if its position changes with time.

    Examples:

    • A car driving from Calgary to Edmonton.
    • A patient walking through a hospital corridor.
    • A conveyor belt moving inventory.

    To describe motion, we need:

    1. Position
    2. Time

    Position

    Position tells us where an object is located relative to a chosen origin.

    Suppose we define a coordinate system:

    -4  -3  -2  -1   0   1   2   3   4

    If a car is at location 3 m, we write

    x = 3 m

    If another car is at location -2 m, we write

    x = -2 m

    The sign tells us the direction relative to the origin.

    Distance

    Distance is the total path traveled.

    Distance ignores direction.

    Example:

    A person walks:

    • 5 m to the right
    • 3 m to the left

    Distance traveled:

    Distance = 5 + 3 = 8 m

    Distance is always positive.

    Displacement

    Displacement measures the change in position.

    The displacement formula is

    [Δx=xf−xi][ \Delta x = x_f – x_i ]

    where:

    • (x_i) = initial position
    • (x_f) = final position

    Example

    Start at:

    xi=2 m x_i = 2 \text{ m}

    End at:

    xf=8 mx_f = 8 \text{ m}

    Displacement:

    Δx=8−2=6 m \Delta x = 8 – 2 = 6 \text{ m}

    Example

    Start at:

    xi=8 m x_i = 8 \text{ m}

    End at:

    xf=2 m x_f = 2 \text{ m}

    Displacement:

    Δx=2−8=−6 m \Delta x = 2 – 8 = -6 \text{ m}

    The negative sign indicates movement in the negative direction.

    Distance vs Displacement

    Suppose a person walks:

    • 10 m east
    • then 10 m west

    Distance:

    10+10=20 m 10 + 10 = 20 \text{ m}

    Displacement:

    0 m 0 \text{ m}

    because the person returned to the starting point.

    Average Speed

    Average speed measures how quickly distance is covered.

    Formula:

    Average Speed=DistanceTime\text{Average Speed}=\frac{\text{Distance}}{\text{Time}}

    Example

    A car travels 100 km in 2 hours.

    Average Speed=1002=50 \text{Average Speed}=\frac{100}{2}=50

    Average speed:

    50 km/h 50 \text{ km/h}

    Velocity

    Velocity measures the rate of change of displacement.

    Formula:

    v=ΔxΔt v=\frac{\Delta x}{\Delta t}

    Velocity includes direction.

    Example

    A person moves 20 m east in 4 s.

    v=204=5 v=\frac{20}{4}=5

    Velocity:

    5 m/s 5 \text{ m/s}

    Example

    A person moves 20 m west in 4 s.

    v=−5 m/s v=-5 \text{ m/s}

    The negative sign indicates westward motion.

    Speed vs Velocity

    Speed ignores direction.

    Velocity includes direction.

    For example:

    Travel 10 km east and then 10 km west.

    Distance:

    20 km 20 \text{ km}

    Displacement:

    0 km 0 \text{ km}

    Average speed is positive.

    Average velocity is zero.

    Instantaneous Velocity

    Average velocity describes motion over a time interval.

    Sometimes we need velocity at a specific instant.

    Examples:

    • A car’s speedometer
    • Blood velocity measured by ultrasound
    • Aircraft speed

    Using calculus:

    v=dxdt v=\frac{dx}{dt}

    This derivative measures the rate of change of position.

    Acceleration

    Acceleration measures how rapidly velocity changes.

    Formula:

    a=ΔvΔt a=\frac{\Delta v}{\Delta t}

    Units:

    m/s2 \text{m/s}^2

    Instantaneous Acceleration

    Using calculus:

    a=dvdt a=\frac{dv}{dt}

    Example

    A car accelerates from

    10 m/s 10 \text{ m/s}

    to

    30 m/s 30 \text{ m/s}

    in

    5 s 5 \text{ s}

    Acceleration:

    a=30−105=4 m/s2 a=\frac{30-10}{5}=4 \text{ m/s}^2

    Position-Time Graphs

    A position-time graph shows how position changes with time.

    Object at Rest

    Horizontal line.

    Position remains constant.

    Velocity:

    v=0 v = 0

    Constant Velocity

    Straight sloping line.

    The slope equals velocity.

    Slope=ΔxΔt=v \text{Slope}=\frac{\Delta x}{\Delta t}=v

    Velocity-Time Graphs

    A velocity-time graph shows how velocity changes with time.

    Constant Velocity

    Horizontal line.

    Acceleration:

    a=0 a = 0

    Constant Acceleration

    Straight sloping line.

    The slope equals acceleration.

    Slope=ΔvΔt=a \text{Slope}=\frac{\Delta v}{\Delta t}=a

    Area Under a Velocity-Time Graph

    The area under a velocity-time graph gives displacement.

    For constant velocity:

    Displacement=vt \text{Displacement}=v t

    Example:

    v=10 m/s v = 10 \text{ m/s}
    t=5 s t = 5 \text{ s}

    Therefore:

    x=50 m x = 50 \text{ m}

    Constant Acceleration Equations

    These equations are among the most important formulas in introductory physics.

    Equation 1

    v=v0+at v=v_0+at

    Equation 2

    x=x0+v0t+12at2 x=x_0 + v_0 t + \frac{1}{2}at^2

    Equation 3

    v2=v02+2a(x−x0) v^2=v_0^2 + 2a(x-x_0)

    Worked Example 1

    A car starts from rest.

    v0=0 v_0 = 0

    Acceleration:

    a=2 m/s2 a = 2 \text{ m/s}^2

    Time:

    t=5 s t = 5 \text{ s}

    Using

    v=v0+at v=v_0+at

    we obtain

    v=0+2(5) v=0+2(5)
    v=10 m/s v=10 \text{ m/s}

    Worked Example 2

    A package starts from rest.

    v0=0 v_0 = 0
    a=1 m/s2a = 1 \text{ m/s}^2
    t=8 s t = 8 \text{ s}

    Using

    x=x0+v0t+12at2 x=x_0 + v_0 t + \frac{1}{2}at^2

    and assuming

    x0=0 x_0 = 0

    gives

    [x=12(1)(8)2][ x=\frac{1}{2}(1)(8)^2 ]
    x=32 m x=32 \text{ m}

    Free Fall

    Objects falling near Earth experience approximately constant acceleration.

    This acceleration is denoted by

    g=9.81 m/s2 g=9.81 \text{ m/s}^2

    directed downward.

    Why Do Heavy and Light Objects Fall Together?

    Newton’s second law:

    F=ma F=ma

    Gravitational force:

    F=mg F=mg

    Combining:

    mg=ma mg=ma

    Canceling mass:

    a=g a=g

    Thus all objects experience the same gravitational acceleration when air resistance is ignored.

    This observation later inspired Einstein’s development of general relativity.

    Healthcare Application

    Blood flow studies often measure:

    • Velocity
    • Acceleration
    • Flow rates

    These measurements help diagnose:

    • Arterial narrowing
    • Valve disease
    • Circulatory disorders

    Supply Chain Application

    Motion concepts appear throughout logistics:

    • Conveyor systems
    • Warehouse robots
    • Automated sorting
    • Vehicle routing

    Velocity and acceleration directly affect throughput and delivery performance.

    Key Takeaways

    1. Motion is a change in position over time.
    2. Distance measures path length.
    3. Displacement measures change in position.
    4. Velocity is the rate of change of position.
    5. Acceleration is the rate of change of velocity.
    6. The slope of a position-time graph equals velocity.
    7. The slope of a velocity-time graph equals acceleration.
    8. The area under a velocity-time graph equals displacement.
    9. Constant acceleration leads to three fundamental equations.
    10. Free fall is motion under constant gravitational acceleration.

    References

    • University Physics
    • Fundamentals of Physics
    • Physics for Scientists and Engineers
    • Galileo Galilei
    • Isaac Newton

    Next, Lesson 3 will introduce vectors and motion in two dimensions, which is the mathematical foundation for forces, electric fields, magnetic fields, quantum states, and eventually relativity.

    Related posts:

    1. Lesson 1: Units, Measurements, and Scientific Thinking
    2. Lesson 3: Motion in Two Dimensions and Vectors
    3. Python 3: Files and Exception Handling
    4. Python 10: Working with APIs and Web Data

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