Motion Diagrams and Average Velocity Ranking Guide

Overview of Motion Diagrams and Strobe Photography

• Flash strobe photography captures the instantaneous position of moving objects at fixed, equal time intervals on a single visual frame.

• In this experiment:

  • Each dot represents the location of a sphere at a specific photograph timestamp, recorded every second ($\Delta t = 1.0\,\text{s}$).

  • All spheres travel along a horizontal axis moving strictly from left to right.

  • Displacements and spatial positions are measured in meters ($\text{m}$).

  • The total observation duration varies per sphere, but the target evaluation interval is strictly the first $3.0\,\text{s}$ of motion ($t = 0.0\,\text{s}$ to $t = 3.0\,\text{s}$).

Core Theoretical Principles and Equations

• Position ($x$):

  • The exact location of a particle along the coordinate line relative to the origin ($x = 0.0\,\text{m}$) at time t$.\n- **Displacement (\Delta x)**:\n - The vector quantity defining the straight-line change in position from an initial time t_i$to a final time$t_f:\n    \Delta x = x(t_f) - x(t_i)\n - For the first 3.0\,\text{s}$interval ($t_i = 0.0\,\text{s}$and$t_f = 3.0\,\text{s}):\n    \Delta x = x(3.0\,\text{s}) - x(0.0\,\text{s})\n- **Average Velocity (v_{\text{avg}})**:\n - The total displacement divided by the time interval required for that displacement:\n    v_{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{x(t_f) - x(t_i)}{t_f - t_i}\n - Substituting \Delta t = 3.0\,\text{s} - 0.0\,\text{s} = 3.0\,\text{s}:\n    v_{\text{avg}} = \frac{x(3.0\,\text{s}) - x(0.0\,\text{s})}{3.0\,\text{s}}\n- **Interpretation of Dot Spacing**:\n - **Uniform spacing**: Constant velocity (a = 0.0\,\text{m/s}^2).\n - **Increasing spacing**: Object is speeding up / accelerating (a > 0.0\,\text{m/s}^2).\n - **Decreasing spacing**: Object is slowing down / decelerating (a < 0.0\,\text{m/s}^2).\n- **Strobe Dot to Timestamp Mapping**:\n - 1st Dot: t = 0.0\,\text{s}\n - 2nd Dot: t = 1.0\,\text{s}\n - 3rd Dot: t = 2.0\,\text{s}\n - 4th Dot: t = 3.0\,\text{s}\n\n# Detailed Individual Sphere Motion Calculations\n\n- **Sphere A**:\n - Position at t = 0.0\,\text{s}$(1st dot):$x_A(0.0\,\text{s}) = 0.0\,\text{m}\n - Position at t = 1.0\,\text{s}$(2nd dot):$x_A(1.0\,\text{s}) = 2.0\,\text{m}\n - Position at t = 2.0\,\text{s}$(3rd dot):$x_A(2.0\,\text{s}) = 4.0\,\text{m}\n - Position at t = 3.0\,\text{s}$(4th dot):$x_A(3.0\,\text{s}) = 6.0\,\text{m}\n - Net displacement over first 3.0\,\text{s}:\n    \Delta x_A = 6.0\,\text{m} - 0.0\,\text{m} = 6.0\,\text{m}\n - Average velocity:\n    v_{\text{avg}, A} = \frac{6.0\,\text{m}}{3.0\,\text{s}} = 2.0\,\text{m/s}\n - Motion type: Uniform, constant velocity (2.0\,\text{m/s}).\n\n- **Sphere B**:\n - Position at t = 0.0\,\text{s}$(1st dot):$x_B(0.0\,\text{s}) = 0.0\,\text{m}\n - Position at t = 1.0\,\text{s}$(2nd dot):$x_B(1.0\,\text{s}) = 1.0\,\text{m}\n - Position at t = 2.0\,\text{s}$(3rd dot):$x_B(2.0\,\text{s}) = 2.0\,\text{m}\n - Position at t = 3.0\,\text{s}$(4th dot):$x_B(3.0\,\text{s}) = 3.0\,\text{m}\n - Net displacement over first 3.0\,\text{s}:\n    \Delta x_B = 3.0\,\text{m} - 0.0\,\text{m} = 3.0\,\text{m}\n - Average velocity:\n    v_{\text{avg}, B} = \frac{3.0\,\text{m}}{3.0\,\text{s}} = 1.0\,\text{m/s}\n - Motion type: Uniform, constant velocity (1.0\,\text{m/s}).\n\n- **Sphere C**:\n - Position at t = 0.0\,\text{s}$(1st dot):$x_C(0.0\,\text{s}) = 0.0\,\text{m}\n - Position at t = 1.0\,\text{s}$(2nd dot):$x_C(1.0\,\text{s}) \approx 0.33\,\text{m}\n - Position at t = 2.0\,\text{s}$(3rd dot):$x_C(2.0\,\text{s}) = 1.0\,\text{m}\n - Position at t = 3.0\,\text{s}$(4th dot):$x_C(3.0\,\text{s}) = 2.0\,\text{m}\n - Net displacement over first 3.0\,\text{s}:\n    \Delta x_C = 2.0\,\text{m} - 0.0\,\text{m} = 2.0\,\text{m}\n - Average velocity:\n    v_{\text{avg}, C} = \frac{2.0\,\text{m}}{3.0\,\text{s}} \approx 0.67\,\text{m/s}\n - Motion type: Non-uniform, accelerating motion.\n\n- **Sphere D**:\n - Position at t = 0.0\,\text{s}$(1st dot):$x_D(0.0\,\text{s}) = 0.0\,\text{m}\n - Position at t = 1.0\,\text{s}$(2nd dot):$x_D(1.0\,\text{s}) = 3.0\,\text{m}\n - Position at t = 2.0\,\text{s}$(3rd dot):$x_D(2.0\,\text{s}) = 6.0\,\text{m}\n - Position at t = 3.0\,\text{s}$(4th dot):$x_D(3.0\,\text{s}) = 9.0\,\text{m}\n - Net displacement over first 3.0\,\text{s}:\n    \Delta x_D = 9.0\,\text{m} - 0.0\,\text{m} = 9.0\,\text{m}\n - Average velocity:\n    v_{\text{avg}, D} = \frac{9.0\,\text{m}}{3.0\,\text{s}} = 3.0\,\text{m/s}\n - Motion type: Uniform, constant velocity (3.0\,\text{m/s}).\n\n- **Sphere E**:\n - Position at t = 0.0\,\text{s}$(1st dot):$x_E(0.0\,\text{s}) = 0.0\,\text{m}\n - Position at t = 1.0\,\text{s}$(2nd dot):$x_E(1.0\,\text{s}) = 4.0\,\text{m}\n - Position at t = 2.0\,\text{s}$(3rd dot):$x_E(2.0\,\text{s}) = 6.0\,\text{m}\n - Position at t = 3.0\,\text{s}$(4th dot):$x_E(3.0\,\text{s}) = 7.0\,\text{m}\n - Net displacement over first 3.0\,\text{s}:\n    \Delta x_E = 7.0\,\text{m} - 0.0\,\text{m} = 7.0\,\text{m}\n - Average velocity:\n    v_{\text{avg}, E} = \frac{7.0\,\text{m}}{3.0\,\text{s}} \approx 2.33\,\text{m/s}\n - Motion type: Non-uniform, decelerating motion.\n\n- **Sphere F**:\n - Position at t = 0.0\,\text{s}$(1st dot):$x_F(0.0\,\text{s}) = 1.0\,\text{m}\n - Position at t = 1.0\,\text{s}$(2nd dot):$x_F(1.0\,\text{s}) = 6.0\,\text{m}\n - Position at t = 2.0\,\text{s}$(3rd dot):$x_F(2.0\,\text{s}) = 9.0\,\text{m}\n - Position at t = 3.0\,\text{s}$(4th dot):$x_F(3.0\,\text{s}) = 10.0\,\text{m}\n - Net displacement over first 3.0\,\text{s}:\n    \Delta x_F = 10.0\,\text{m} - 1.0\,\text{m} = 9.0\,\text{m}\n - Average velocity:\n    v_{\text{avg}, F} = \frac{9.0\,\text{m}}{3.0\,\text{s}} = 3.0\,\text{m/s}\n - Motion type: Non-uniform, decelerating motion starting from initial offset position x_0 = 1.0\,\text{m}.\n\n# Complete Ranking and Summary of Average Velocities\n\n- **Calculated Average Velocities (t = 0.0\,\text{s}$to$t = 3.0\,\text{s})**:\n - Sphere D: 3.0\,\text{m/s}\n - Sphere F: 3.0\,\text{m/s}\n - Sphere E: \approx 2.33\,\text{m/s}\n - Sphere A: 2.0\,\text{m/s}\n - Sphere B: 1.0\,\text{m/s}\n - Sphere C: \approx 0.67\,\text{m/s}\n\n- **Ranking from Greatest (Rank 1) to Least (Rank 5)**:\n - **Rank 1 (Greatest)**: Sphere D, Sphere F (Tied)\n - **Rank 2**: Sphere E\n - **Rank 3**: Sphere A\n - **Rank 4**: Sphere B\n - **Rank 5 (Least)**: Sphere C\n\n- **Mathematical Ordering Relationship**:\n  v_{\text{avg}, D} = v_{\text{avg}, F} > v_{\text{avg}, E} > v_{\text{avg}, A} > v_{\text{avg}, B} > v_{\text{avg}, C}\n\n# Critical Conceptual Distinctions and Problem-Solving Nuances\n\n- **Distinguishing Position vs. Displacement**:\n - Sphere F reaches position x = 10.0\,\text{m}$at$t = 3.0\,\text{s}$, whereas Sphere D reaches position$x = 9.0\,\text{m}.\n - Despite ending further down the track, Sphere F began at x = 1.0\,\text{m}$, so its total displacement is$\Delta x_F = 10.0\,\text{m} - 1.0\,\text{m} = 9.0\,\text{m}.\n - Average velocity depends strictly on net displacement \Delta x$, not final position$x_f.\n- **Proper Counting of Strobe Intervals**:\n - A duration of 3.0\,\text{s}$consists of 3 full time steps of$1.0\,\text{s} each.\n - Because the initial state at t = 0.0\,\text{s}$is represented by the 1st dot, the state at$t = 3.0\,\text{s}$$ is represented by the 4th dot (not the 3rd dot).

• Effect of Non-Uniform Motion on Interval Analysis:

  • Average velocity reflects overall net displacement divided by total time regardless of whether the speed was constant, increasing, or decreasing within that timeframe.