Visualizing Velocity: Position-Time Graphs and One-Dimensional Motion

Displacement and One-Dimensional Velocity

• Position and Displacement Fundamentals:

  • Displacement represents the net change in position, defined mathematically as:         $\Delta x = x_{\text{final}} - x_{\text{initial}}$

  • Displacement accounts for both direction and distance, whereas total distance traveled accounts only for the overall path length covered.

  • Example calculation on a horizontal line:

    • First leg of motion: Move $6$ positions to the right ($+6\,\text{miles}$).

    • Second leg of motion: Move $4$ positions to the left ($-4\,\text{miles}$).

    • Third leg of motion: Move $6$ positions to the left ($-6\,\text{miles}$).

    • Total displacement calculation:             $\Delta x = (+6\,\text{miles}) + (-4\,\text{miles}) + (-6\,\text{miles}) = -4\,\text{miles}$

    • The positive movement of $+6\,\text{miles}$ and negative movement of $-6\,\text{miles}$ cancel each other out, yielding a net change in position of $-4\,\text{miles}$ (a net motion of $4\,\text{miles}$ to the left).

• Velocity Calculation from Displacement:

  • Average velocity ($v$) is defined as the change in position divided by the total time elapsed ($\Delta t$):         $v = \frac{\Delta x}{\Delta t}$

  • Evaluating velocity across individual legs of a journey, assuming each segment takes a duration of $\Delta t = 2\,\text{minutes}$:

    • Trip 1: Traveled $+6\,\text{miles}$ in $2\,\text{minutes}$:             $v_1 = \frac{+6\,\text{miles}}{2\,\text{minutes}} = +3\,\text{miles\,min}^{-1}$             This represents rapid forward motion (motion to the right).

    • Trip 2: Traveled $-4\,\text{miles}$ in $2\,\text{minutes}$:             $v_2 = \frac{-4\,\text{miles}}{2\,\text{minutes}} = -2\,\text{miles\,min}^{-1}$             This represents slower backward motion (motion to the left).

    • Trip 3: Traveled $-6\,\text{miles}$ in $2\,\text{minutes}$:             $v_3 = \frac{-6\,\text{miles}}{2\,\text{minutes}} = -3\,\text{miles\,min}^{-1}$             This represents faster backward motion (motion to the left).

  • Each velocity value explicitly combines speed (magnitude) and direction (sign).

• Defining the Origin on a Number Line:

  • The origin ($x = 0$) serves as the reference point for measuring position.

  • The origin can be set at $0$ or designated at the starting location of an object (e.g., a car, bus, truck, or rat).

  • Movement to the right relative to the origin corresponds to positive motion ($+$).

  • Movement to the left relative to the origin corresponds to negative motion ($-$).

Consistent Origin and Directional Motion

• Tracking Position Relative to a Fixed Origin:

  • Establishing a consistent origin allows for clear categorization of motion toward or away from the reference point.

  • Consider an object with an initial position at $x = 100\,\text{meters}$ relative to an origin:

    • Position $100\,\text{m} \rightarrow 95\,\text{m}$: Displacement $\Delta x = 95\,\text{m} - 100\,\text{m} = -5\,\text{m}$. The object is moving toward the origin.

    • Position $95\,\text{m} \rightarrow 70\,\text{m}$ (over $2\,\text{seconds}$): Displacement $\Delta x = 70\,\text{m} - 95\,\text{m} = -25\,\text{m}$. The object continues moving toward the origin.

    • Position $70\,\text{m} \rightarrow 40\,\text{m}$: Displacement $\Delta x = 40\,\text{m} - 70\,\text{m} = -30\,\text{m}$. The object is moving toward the origin.

    • Position $40\,\text{m} \rightarrow 20\,\text{m}$: Displacement $\Delta x = 20\,\text{m} - 40\,\text{m} = -20\,\text{m}$. The object is moving toward the origin.

    • Position $20\,\text{m} \rightarrow 20\,\text{m}$: Displacement $\Delta x = 20\,\text{m} - 20\,\text{m} = 0\,\text{m}$. The position remains unchanged; therefore, velocity is $0\,\text{m/s}$ and there is no motion.

    • Position $20\,\text{m} \rightarrow 40\,\text{m}$: Displacement $\Delta x = 40\,\text{m} - 20\,\text{m} = +20\,\text{m}$. The object is moving away from the origin.

    • Position $40\,\text{m} \rightarrow 70\,\text{m}$: Displacement $\Delta x = 70\,\text{m} - 40\,\text{m} = +30\,\text{m}$. The object is moving away from the origin.

• Sign-Direction Relationship:

  • Negative displacement ($\Delta x < 0$) or decreasing distance to the origin indicates motion toward the origin.

  • Positive displacement ($\Delta x > 0$) or increasing distance from the origin indicates motion away from the origin.

Graphical Representation of Motion and Cart Experiments

• Experimental Setup for Data Collection:

  • Motion data is collected using a dynamics cart positioned on a track or ramp system.

  • The cart is equipped with integrated sensors to track position, acceleration, velocity, and force.

  • Data is transmitted via Bluetooth in real time to a computer analysis program.

  • An origin point ($x = 0$) is designated at one end of the ramp, establishing a fixed positive direction ($+x$) directed away from the origin.

• Position vs. Time Graph Characteristics:

  • Moving away from the origin: Produces a positive slope on a position vs. time graph.

  • Moving toward the origin: Produces a negative slope on a position vs. time graph.

  • Stationary state: Produces a flat, horizontal line (zero slope) on a position vs. time graph, corresponding to zero velocity.

• Comparing and Contrasting Graph Slopes:

  • Comparison (Similarities): Slopes that display equal steepness (angles) possess an equal magnitude of slope. This indicates that the object moves at the same speed during both intervals.

  • Contrast (Differences): A positive slope denotes movement directed away from the origin, while a negative slope denotes movement directed toward the origin. Thus, equal steepness with opposite signs reflects equal speed in opposite directions.

Calculating Velocity from Graph Slopes

• Slope Formula:

  • The slope ($m$) of a line on a position-time graph corresponds directly to velocity ($v$):         $v = \text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

  • Where $y$ represents position in meters ($\text{m}$) and $x$ represents time in seconds ($\text{s}$).

• Calculation 1: Positive Slope (Motion Away from Origin):

  • Data points: Point 1 at $(x_1, y_1) = (5\,\text{s}, 0.49\,\text{m})$, Point 2 at $(x_2, y_2) = (6\,\text{s}, 0.64\,\text{m})$.

  • Velocity calculation:         $v = \frac{0.64\,\text{m} - 0.49\,\text{m}}{6\,\text{s} - 5\,\text{s}} = \frac{0.15\,\text{m}}{1\,\text{s}} = +0.15\,\text{m\,s}^{-1}$

  • The positive result confirms movement away from the origin at a rate of $0.15\,\text{m\,s}^{-1}$.

• Calculation 2: Negative Slope (Motion Toward Origin):

  • Data points: Point 1 at $(x_1, y_1) = (12\,\text{s}, 0.64\,\text{m})$, Point 2 at $(x_2, y_2) = (13\,\text{s}, 0.49\,\text{m})$.

  • Velocity calculation:         $v = \frac{0.49\,\text{m} - 0.64\,\text{m}}{13\,\text{s} - 12\,\text{s}} = \frac{-0.15\,\text{m}}{1\,\text{s}} = -0.15\,\text{m\,s}^{-1}$

  • The negative result confirms movement toward the origin at a rate of $0.15\,\text{m\,s}^{-1}$.

  • Comparing both calculations demonstrates equal speed ($0.15\,\text{m\,s}^{-1}$) in opposing directions.

• Slope Intensity and Speed:

  • Slope intensity (steepness) corresponds directly to the magnitude of velocity (speed).

  • A less intense (flatter) slope represents a smaller change in position over time, indicating a lower velocity (slower speed).

  • A more intense (steeper) slope represents a larger change in position over time, indicating a higher velocity (faster speed).

Interpreting Complex Position-Time Graphs

• Key Graph Features and Interpretations:

  • Zero Velocity Segments (Horizontal Lines):

    • Any flat segment where the slope is zero ($\text{slope} = 0$) indicates zero change in position over time.

    • The object is stationary ($v = 0\,\text{m\,s}^{-1}$).

    • In multi-segment motion graphs, multiple flat regions denote separate periods where the object remains at rest (e.g., three distinct stationary periods).

  • Positive Slope Segments (Motion Away):

    • Lines sloped upward from left to right indicate positive velocity ($v > 0$), meaning motion away from the origin.

    • Comparing two positive slopes: A gentler slope represents a slower movement away, whereas a steeper slope represents a faster movement away.

  • Negative Slope Segments (Motion Toward):

    • Lines sloped downward from left to right indicate negative velocity ($v < 0$), meaning motion toward the origin.

    • A extended segment with a shallow downward slope indicates slow motion toward the origin over a longer time interval.

• Summary of Graph Interpretation Capabilities:

  • Direction of motion is determined by the sign of the slope (positive = away, negative = toward).

  • State of motion is determined by whether a slope exists (horizontal line = zero velocity/stationary).

  • Relative speed is determined by the steepness/intensity of the slope (steeper = faster, flatter = slower).

  • Exact quantitative velocity is determined by calculating the numerical slope $\frac{\Delta x}{\Delta t}$ using coordinate data points.