AP Q1 Class 3 - Acceleration - 2026-27

Motion Terminology: Vectors and Scalars

• Scalar Quantities

  • A scalar quantity is defined by having a magnitude, which consists of a number and units.
  • Scalar quantities do not possess a direction.
  • Examples of scalar quantities include:
    • Distance ($d$)
    • Speed ($v$)

• Vector Quantities

  • A vector quantity is defined by having both a magnitude and a specific direction.
  • Examples of vector quantities include:
    • Displacement ($\Delta \vec{x}$)
    • Velocity ($\vec{v}$)

Average Speed and Average Velocity

• Average Speed ($v_{avg}$)

  • Average speed is calculated as the total distance traveled over the change in time.
  • Formula: $v_{avg} = \frac{d}{\Delta t}$

• Average Velocity ($\vec{v}_{avg}$)

  • Average velocity is calculated as the displacement over the change in time.
  • Formula: $\vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t}$

• Key Distinction

  • It is critical to note that average speed and average velocity can result in completely different numerical values depending on the path taken.

Linear Graphing Fundamentals

• Linear Equations
  • A linear relationship on a graph is represented by the formula: $y = mx + b$
  • Intercept ($b$): The point where the line crosses the y-axis.
  • Slope ($m$): Represents the steepness and direction of the line.
    • Formula: $m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}$

Analyzing Motion with Graphs

• Position vs. Time Graph ($x-t$)

  • A straight (linear) line on a position-time graph indicates motion with constant velocity.
  • If the velocity were changing, the graph would appear curved rather than a straight line.
  • The slope of a position-time graph is equal to the velocity ($\vec{v}$).

• Velocity vs. Time Graph ($v-t$)

  • This graph plots velocity ($v$ in $m/s$) against time ($t$ in $s$).
  • Slope: The slope of a velocity-time graph represents the acceleration ($\vec{a}$).
  • Example Case: Jet Car Velocity
    • The graph shows a linear increase in velocity over time.
    • Time scale: $0$ to $40\,s$.
    • Velocity scale: up to $180\,m/s$.
    • The line begins at the origin ($0,0$) and reaches approximately $160\,m/s$ at $40\,s$.

Detailed Motion Terminology and Kinematic Definitions

• Position ($\vec{x}$)

  • Definition: The specific location of an object.
  • Standard Units: meters ($m$).

• Velocity ($\vec{v}$)

  • Definition: The rate of change of position with respect to time.
  • Standard Units: meters per second ($m/s$).
  • Formula: $\vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t}$

• Acceleration ($\vec{a}$)

  • Definition: The rate of change of velocity with respect to time.
  • Standard Units: meters per second squared ($m/s^2$).
  • Formula: $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}$

Mathematical Relationships in Constant Acceleration Graphs

• Slopes

  • The slope of a Position vs. Time graph yields Velocity.
  • The slope of a Velocity vs. Time graph yields Acceleration.

• Areas Under the Curve

  • The area under a Velocity vs. Time graph represents the Change in Position (Displacement).
  • The area under an Acceleration vs. Time graph represents the Change in Velocity.

Directionality of Vectors

• Representing Direction

  • The direction of a vector can be represented using specific words (e.g., left, right, up, down, North, South) or mathematically through the sign ($+$ or $-$) of the vector.

• Direction Interpretations

  • Direction of Position: Indicates the location of the object relative to a defined origin.
  • Direction of Velocity: Indicates the direction in which the object is moving.
  • Direction of Acceleration: Indicates the direction in which the velocity is changing.

Determining the Direction of Acceleration

• Speeding Up vs. Slowing Down

  • Speeding Up: Occurs when the acceleration is in the same direction as the velocity.
  • Slowing Down: Occurs when the acceleration is in the opposite direction to the velocity.

• Applied Examples

  1. Scenario: A car is driving to the right and speeding up.
     • Velocity direction: Right.
     • Acceleration direction: Right (matching velocity).
  2. Scenario: A car is driving to the left and speeding up.
     • Velocity direction: Left.
     • Acceleration direction: Left (matching velocity).
  3. Scenario: A car is driving to the left and slowing down.
     • Velocity direction: Left.
     • Acceleration direction: Right (opposite to velocity).

Precision Graphing Techniques

• Labeling: Axes must always be labeled with both the quantity and the appropriate units (e.g., "$v\, (m/s)$").
• Scaling: Axes should be scaled so that the plotted data covers more than half of the available space on each axis.
• Best Fit Line:
  • Must be a single straight line drawn with a straight edge.
  • There should be approximately half of the data points above the line and half below the line.
• Slope Calculation:
  • When calculating the slope from