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: $\Delta x = x<em>f - x</em>i$ or equivalently $\Delta x = x - x_0\,,$ where:
  • $x<em>i$ or $x</em>0$ 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: $\Delta x = x<em>f - x</em>i = 4.0\,\text{m} - 2.0\,\text{m} = 2.0\,\text{m}$ (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).