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.}