Distance vs. Time Activity

Activity Overview and Graphing Directions

Activity Type and Context:
- The document serves as a structured physical science worksheet designed to instruct students on the principles of 1-dimensional kinematics, specifically constructing and analyzing a distance-time graph.
- Header fields include designated sections for student identification: Name and Date.
Step-by-Step Task Instructions:
- Data Extraction: Read and interpret the paired coordinate values provided in the data table.
- Axis Labeling: Explicitly assign and label both the independent variable and dependent variable along their appropriate coordinate axes.
- Scale Selection: Establish an appropriate, uniform, and proportional numerical scale across each axis to span the entire domain and range without distortion.
- Coordinate Plotting: Accurately plot each ordered pair $(t, d)$ as discrete data points on the Cartesian coordinate grid.
- Line Construction: Connect the plotted points sequentially with a continuous straight line to demonstrate the functional relationship between time and distance.

Distance vs. Time activity worksheet showing instructions, kinematic data table, and coordinate grid

Kinematic Data and Coordinate Pairs

Tabular Data Values:
- The provided data records an object's position over time starting from rest at the origin ( $0\,\text{s}$ , $0\,\text{m}$ ):
  - At time $t = 0\,\text{s}$ , distance $d = 0\,\text{m}$
  - At time $t = 1\,\text{s}$ , distance $d = 5\,\text{m}$
  - At time $t = 2\,\text{s}$ , distance $d = 10\,\text{m}$
  - At time $t = 3\,\text{s}$ , distance $d = 15\,\text{m}$
  - At time $t = 4\,\text{s}$ , distance $d = 20\,\text{m}$
  - At time $t = 5\,\text{s}$ , distance $d = 25\,\text{m}$
Cartesian Coordinate Mapping:
- Each row represents an ordered pair of coordinates $(x, y)$ corresponding to $(t, d)$ :
  - $\text{Point } 1: (0, 0)$
  - $\text{Point } 2: (1, 5)$
  - $\text{Point } 3: (2, 10)$
  - $\text{Point } 4: (3, 15)$
  - $\text{Point } 5: (4, 20)$
  - $\text{Point } 6: (5, 25)$

Graph Construction and Grid Specifications

Horizontal Axis (x-axis) Configuration:
- Physical Quantity: Time ( $t$ ), representing the independent variable.
- Standard Unit: Seconds ( $\text{s}$ ).
- Axis Label: "Time (s)".
- Domain: $[0, 5]\,\text{s}$ .
- Major Numerical Intervals: Displayed at integer increments of $1\,\text{s}$ ( $0$ , $1$ , $2$ , $3$ , $4$ , $5$ ).
- Subdivision Resolution: Two subdivisions per major unit, yielding a grid resolution of $0.5\,\text{s}$ per grid square.
Vertical Axis (y-axis) Configuration:
- Physical Quantity: Distance ( $d$ ), representing the dependent variable.
- Standard Unit: Meters ( $\text{m}$ ).
- Axis Label: "Distance (m)".
- Range: $[0, 25]\,\text{m}$ .
- Major Numerical Intervals: Displayed at increments of $5\,\text{m}$ ( $0$ , $5$ , $10$ , $15$ , $20$ , $25$ ).
- Subdivision Resolution: Five grid subdivisions between each major increment of $5\,\text{m}$ , indicating that each individual vertical grid square corresponds exactly to $1\,\text{m}$ .

Mathematical Analysis and Velocity Calculations

Slope Definition on a Distance-Time Graph:
- In kinematics, the gradient (slope) of a position-time or distance-time curve is defined as the rate of change of position with respect to time, which equals speed (or magnitude of velocity):
- $\text{Slope} = \frac{\Delta y}{\Delta x} = \frac{\Delta d}{\Delta t} = \text{Speed } (v)$
Interval-by-Interval Velocity Determination:
- Interval 1 ( $0\,\text{s}$ to $1\,\text{s}$ ):
  - $v_1 = \frac{5\,\text{m} - 0\,\text{m}}{1\,\text{s} - 0\,\text{s}} = \frac{5\,\text{m}}{1\,\text{s}} = 5\,\text{m}\,\text{s}^{-1}$
- Interval 2 ( $1\,\text{s}$ to $2\,\text{s}$ ):
  - $v_2 = \frac{10\,\text{m} - 5\,\text{m}}{2\,\text{s} - 1\,\text{s}} = \frac{5\,\text{m}}{1\,\text{s}} = 5\,\text{m}\,\text{s}^{-1}$
- Interval 3 ( $2\,\text{s}$ to $3\,\text{s}$ ):
  - $v_3 = \frac{15\,\text{m} - 10\,\text{m}}{3\,\text{s} - 2\,\text{s}} = \frac{5\,\text{m}}{1\,\text{s}} = 5\,\text{m}\,\text{s}^{-1}$
- Interval 4 ( $3\,\text{s}$ to $4\,\text{s}$ ):
  - $v_4 = \frac{20\,\text{m} - 15\,\text{m}}{4\,\text{s} - 3\,\text{s}} = \frac{5\,\text{m}}{1\,\text{s}} = 5\,\text{m}\,\text{s}^{-1}$
- Interval 5 ( $4\,\text{s}$ to $5\,\text{s}$ ):
  - $v_5 = \frac{25\,\text{m} - 20\,\text{m}}{5\,\text{s} - 4\,\text{s}} = \frac{5\,\text{m}}{1\,\text{s}} = 5\,\text{m}\,\text{s}^{-1}$
Total Average Velocity:
- $v_{\text{avg}} = \frac{d_{\text{total}}}{t_{\text{total}}} = \frac{25\,\text{m} - 0\,\text{m}}{5\,\text{s} - 0\,\text{s}} = 5\,\text{m}\,\text{s}^{-1}$
Linear Functional Equation:
- Because the line passes directly through the origin $(0, 0)$ with a constant slope $m = 5\,\text{m}\,\text{s}^{-1}$ , the equation modeling the motion is:
- $d(t) = 5t$
- where $d$ is in meters and $t$ is in seconds.

Kinematic Interpretation and Motion Principles

Uniform Motion (Constant Speed):
- A linear distance-time graph featuring a constant, non-zero slope represents constant speed (uniform velocity).
- Equal changes in distance ( $\Delta d = 5\,\text{m}$ ) occur over equal increments of time ( $\Delta t = 1\,\text{s}$ ).
Acceleration Profile:
- Acceleration ( $a$ ) is defined as the time derivative of velocity:
- $a = \frac{dv}{dt} = \frac{\Delta v}{\Delta t}$
- Because velocity is invariant across all intervals ( $v(t) = 5\,\text{m}\,\text{s}^{-1}$ ),
- $a = 0\,\text{m}\,\text{s}^{-2}$
- Zero curvature (absence of parabolic or higher-order trajectory) on a distance-time graph confirms the absence of net acceleration.
Summary of Distance-Time Graph Slopes:
- Straight horizontal line ( $\text{slope} = 0$ ): Object is stationary/at rest ( $v = 0\,\text{m}\,\text{s}^{-1}$ ).
- Straight inclined line ( $\text{slope} = \text{constant} > 0$ ): Object is traveling forward at a constant speed.
- Curved line sloping upwards (concave up): Object is accelerating (speed is increasing over time).
- Curved line leveling off (concave down): Object is decelerating (speed is decreasing over time).