Chapter 2: Kinematics – One-Dimensional Motion

Kinematics: One-Dimensional Motion

  • Physics has different branches (mechanics, thermodynamics, electromagnetism, optics). This course focuses on mechanics, specifically the sub-branch of Kinematics.
  • Kinematics vs Dynamics:
    • Kinematics: the study of motion without considering causes.
    • Dynamics: the study of motion with consideration of causes.
  • In this chapter, we study Kinematics (motion in one dimension).

Vectors, Scalars, and Coordinate System

  • A vector is a quantity with both magnitude and direction.
  • A scalar is a quantity with only magnitude (no direction).
  • Coordinate system is required to describe the direction of a vector within a reference frame.
  • For one-dimensional motion, use a simple one-dimensional coordinate line.
  • Direction conventions (commonly used in horizontal motion):
    • Right (positive) and left (negative).
  • For vertical motion: up (positive) and down (negative).
  • Reference frame example: Earth is often used as the reference frame.

Position and Reference Frame

  • Position describes where an object is at a given time relative to a reference frame.
  • We commonly reference position to Earth.
  • See Figure 2 (reference frame concept).

Displacement

  • Displacement is the change in position of an object and is a vector (has direction).
  • Formula: \Delta x = xf - xi or equivalently \Delta x = x - x_0\,, where:
    • $xi$ or $x0$ is the initial position,
    • $x_f$ is the final position.
  • Displacement has a direction (to the right/up is positive in the chosen frame, to the left/down is negative).
  • Example 1:
    • Initial position: $x_0 = 2.0\,\text{m}$, final position: $x = 4.0\,\text{m}$
    • Displacement: \Delta x = 4.0\,\text{m} - 2.0\,\text{m} = 2.0\,\text{m} (to the right).
  • Example 2:
    • Initial: $x_0 = 4.0\,\text{m}$, final: $x = 2.0\,\text{m}$
    • Displacement: \Delta x = 2.0\,\text{m} - 4.0\,\text{m} = -2.0\,\text{m} (to the left).
  • Example 3 (path described, final position is 4.0 m):
    • Start: $x_0 = 2.0\,\text{m}$ → move to $x=4.0\,\text{m}$, then to $x=0.0\,\text{m}$, then back to $x=4.0\,\text{m}$.
    • Displacement: \Delta x = xf - xi = 4.0\,\text{m} - 2.0\,\text{m} = 2.0\,\text{m} (to the right).

Distance vs Displacement

  • Distance:
    • The magnitude (size) of displacement.
    • Distance is a scalar (has magnitude only, no direction).
  • Example 4:
    • Initial: $x_0 = 2.0\,\text{m}$, final: $x = 4.0\,\text{m}$
    • Magnitude of displacement: |\Delta x| = |4.0 - 2.0| = 2.0\,\text{m}.
  • Example 5:
    • Initial: $x_0 = 4.0\,\text{m}$, final: $x = 2.0\,\text{m}$
    • Magnitude of displacement: |\Delta x| = |2.0 - 4.0| = 2.0\,\text{m}.$
  • Distance traveled (total path length): scalar; magnitude of the entire path, not just net change.

Distance Traveled Example (Path 2 m → 4 m → 0 m → 4 m)

  • Path segments: $2.0\,\text{m} \rightarrow 4.0\,\text{m}$ (2.0 m), $4.0\,\text{m} \rightarrow 0.0\,\text{m}$ (4.0 m), $0.0\,\text{m} \rightarrow 4.0\,\text{m}$ (4.0 m).
  • Total distance traveled: d = 2.0\,\text{m} + 4.0\,\text{m} + 4.0\,\text{m} = 10.0\,\text{m}.
  • Note: Displacement for this path is still \Delta x = 2.0\,\text{m}.

Time and Elapsed Time

  • Time: A measure of change; time is the interval over which change occurs.
  • Elapsed time (time interval):
    \Delta t = tf - ti
    or simply \Delta t = t - t_0
  • Time provides the scale for rates (velocity, acceleration).

Average Velocity

  • Definition: displacement divided by the time of travel.
  • Vector quantity; its direction matches the displacement direction.
  • Formula:
    \vec{v}{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{xf - xi}{tf - t_i}
  • SI unit: \text{m/s}

Instantaneous Velocity

  • Definition: velocity at a specific instant in time.
  • Vector quantity.
  • Description (from Figure): a point in time $ti$ with velocity $\vec{v}i$ and at a later time $tf$ velocity $\vec{v}f$.
  • Notation commonly represented with the time indices shown in the figure (e.g., $vi$, $vf$ at $ti$, $tf$).

Speed vs Velocity

  • Speed vs velocity distinction:
    • Speed is the magnitude of velocity; a scalar.
    • Velocity is a vector (has both magnitude and direction).
  • Instantaneous speed is the magnitude of instantaneous velocity:
    v = |\vec{v}|\,.
  • Average speed:
    \bar{v} = \frac{d}{\Delta t}\,,
    where $d$ is the total distance traveled.
  • SI unit for speed: \text{m/s}

Average Acceleration

  • Definition: the rate at which velocity changes.
  • Acceleration is a vector; points in the same direction as the change in velocity.
  • Since velocity is a vector, acceleration can change due to changes in magnitude or direction (or both).
  • Formula:
    \vec{a}{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v} - \vec{v}0}{t - t_0}\,.
  • SI unit: \text{m}/\text{s}^2

Constant Acceleration

  • In this course, we study motion with constant acceleration.
  • Therefore, the average acceleration is a constant acceleration: \vec{a}_{\text{avg}} = \vec{a} = \text{constant} (often written simply as $a = \Delta v/\Delta t$).
  • Sign conventions:
    • If the object speeds up, the acceleration is positive: a > 0.
    • If the object slows down (decelerates), the acceleration is negative: a < 0$$.
  • Key implications: with constant acceleration, velocity changes linearly with time, and displacement depends on both $v_0$, $a$, and $t$ (not shown in transcript but commonly follows in subsequent chapters).