Study Notes on Velocity

1. Definition of Velocity

• Intuitive Understanding

  • We have an intuitive sense of relative speeds (fast vs. slow).

• Precision Through Motion Diagrams

  • Examine motion diagrams of objects moving at a constant speed in a straight line.

  • Motion at constant speed is termed uniform motion.

  • Motion diagrams for uniform motion show equally spaced positions in successive frames, indicating consistent displacement.

  • Example: Skateboarder in Section 1.1 illustrates uniform motion with equal displacements between frames.

2. Motion Diagrams Comparison

• Displacement and Speed Relation

  • Example: Motion diagrams of a bicycle and a car traveling on the same street (as shown in FIGURE 1.16).

  • Observation: The car moves faster than the bicycle.

  • Displacement values:

  • Car: $\Delta x = 40 \text{ ft}$ in 1-second interval.

  • Bicycle: $\Delta x = 20 \text{ ft}$ in 1-second interval.

2.1 Speed Definition

• Speed Definition:
  $\text{speed} = \frac{\text{distance traveled in a given time interval}}{\text{time interval}}$

• Calculating Speed:

  • For bicycle:
    $\text{speed} = \frac{20 \text{ ft}}{1 \text{ s}} = 20 \text{ ft/s}$

  • For car:
    $\text{speed} = \frac{40 \text{ ft}}{1 \text{ s}} = 40 \text{ ft/s}$

• Relative Speed:

  • The car's speed is twice that of the bicycle.

• Units: The unit of speed is feet per second ($\text{ft/s}$), analogous to miles per hour ($\text{mi/h}$).

3. Characterization of Motion

• Velocity Definition: To fully characterize an object's motion, both speed and direction must be specified.

• Comparison of Two Bicycles:

  • FIGURE 1.17 shows two bicycles traveling at the same speed of $20 \text{ ft/s}$ but in different directions.

3.1 Velocity Calculation

• Velocity of Bicycle 1:

  • Using the 1-second interval from $t = 0 \text{ s}$ to $t = 1 \text{ s}$:
    $\text{velocity} = \frac{\Delta x}{\Delta t} = \frac{x(1) - x(0)}{t(1) - t(0)} = \frac{20 \text{ ft} - 0 \text{ ft}}{1 \text{ s} - 0 \text{ s}} = +20 \text{ ft/s}$

• Velocity of Bicycle 2:

  • Same time interval:
    $\text{velocity} = \frac{\Delta x}{\Delta t} = \frac{x(1) - x(0)}{t(1) - t(0)} = \frac{100 \text{ ft} - 120 \text{ ft}}{1 \text{ s} - 0 \text{ s}} = -20 \text{ ft/s}$

3.2 Positive and Negative Velocity

• Significance of Velocity's Sign:

  • Positive velocity indicates motion to the right (or upward in vertical motion).

  • Negative velocity indicates motion to the left (or downward).

• Important Conceptual Distinction:

  • Speed is always positive.

  • Velocity can be positive or negative; it reflects direction.

4. Average vs. Instantaneous Velocity

• Average Velocity:

  • As derived from Equation 1.2:

  • Represents the mean velocity over defined intervals, e.g., for Bicycle 1 is $20 \text{ ft}$ over a second.

  • Acknowledgment that speed may vary within the time frame, thus average velocity does not guarantee constant speed.

• Concept Development in Future Content:

  • Chapter 2 will expand on instantaneous velocity, emphasizing the object's speed at a specific instant.

5. Units of Speed and Velocity

• Measurement Units:

  • Distance unit (e.g., feet, meters, miles) divided by a time unit (e.g., seconds, hours).

• Common velocity units:

  • Miles per hour ($\text{mi/h}$, pronounced "miles per hour")

  • Meters per second ($\text{m/s}$, pronounced "meters per second").

5.1 Interpretation of 'Per'

• The term 'per' relates the numerator to the denominator:

  • For example, a speed of $23 \text{ m/s}$ implies:
    $\text{Distance} = 23 \text{ meters for each } 1 \text{ second of elapsed time.}$

• Similar use of 'per' seen in other ratios, such as density, with gold density being $19.3 \text{ g/cm}^3$ (grams per cubic centimeter), meaning:
  $19.3 \text{ grams of gold for each cubic centimeter of the metal.}$