Impulse–Momentum Analysis of a Soccer Ball Collision

Momentum Principle (Impulse–Momentum Theorem)

• The impulse–momentum theorem states that the change in momentum equals the impulse delivered to the object:
  $\Delta \mathbf{p} = \mathbf{J} = \mathbf{F}_{\text{net}}\, \Delta t.$
• Momentum is defined as the product of mass and velocity for the problems in this course: $\mathbf{p} = m \mathbf{v}.$
• If the net force is approximately constant over a short interaction interval, the impulse can be written as $\Delta \mathbf{p} = \mathbf{F}<em>{\text{net}}\,\Delta t.$ Hence the net force during the interval is $\mathbf{F}</em>{\text{net}} = \frac{\Delta \mathbf{p}}{\Delta t}.$
• Practical strategy for brief, significant interactions:
  • Observe the object before and after the interaction to determine its initial and final momentum.
  • Estimate the time interval $\Delta t$ during which the interaction (and thus a nonzero net force) occurs.
  • Use the momentum change and $\Delta t$ to estimate the average net force in that interval.
• Important caveat: the force during the interaction could vary with time; using an average $\mathbf{F}_{\text{net}}$ assumes it is roughly constant over $\Delta t$ for modeling purposes.
• This approach is illustrated with a soccer ball example where a touch reverses the ball’s direction.

Example context: Soccer ball collision analyzed from video

• The video analysis shows the ball moving in two constant-velocity regimes, separated by a brief interval where velocity changes due to a nonzero net force from surroundings.
• Data source: position vs time from Tracker software; time between video frames is $\Delta t_{\text{frame}} = \frac{1}{30}\ \text{s}.$
• Assumption about the interaction window: the interaction starts one frame before the velocity peak and ends one frame after the peak, giving a net interaction interval
  $\Delta t = \frac{1}{15}\ \text{s}.$
• The magnitude of the net force is computed from the observed momentum change over this interval (see sections on momentum estimation).

How we estimate momentum before and after the interaction

• Initial momentum estimation (before the peak): use two circled position–time points that bracket the start of the interaction.
  • Points are $(t0,x0)$ and $(t1,x1)$ with $t0 < t1$.
  • Average velocity over the interval is approximated as
    $\mathbf{V}<em>{!i} \approx \frac{x</em>1 - x<em>0}{t</em>1 - t_0}.$
  • Initial momentum:
    $\mathbf{P}<em>{!i} = m \mathbf{V}</em>{!i}.$
  • The momentum direction is along +X, consistent with the chosen coordinate system (plus X points to the right).
• Final momentum estimation (after the peak): use two circled points after the peak, $(t2,x2)$ and $(t3,x3)$.
  • Final velocity:
    $\mathbf{V}<em>{!f} \approx \frac{x</em>3 - x<em>2}{t</em>3 - t_2}.$
  • Final momentum:
    $\mathbf{P}<em>{!f} = m \mathbf{V}</em>{!f}.$
• Notes:
  • The mass $m$ is the soccer ball’s mass used in the model (a given constant in the setup).
  • The direction of final momentum can be opposite to the initial direction if the ball reverses, as observed.

Net impulse and the net force during the interaction

• Change in momentum (impulse) over the interaction:
  $\Delta \mathbf{p} = \mathbf{P}<em>{!f} - \mathbf{P}</em>{!i}.$
• With the interaction duration $\Delta t = \frac{1}{15}\ \text{s},$ the average net force is
  $\mathbf{F}_{\text{net}} \approx \frac{\Delta \mathbf{p}}{\Delta t}.$
• Example result from the transcript: the estimated net force is
  $\mathbf{F}_{\text{net}} = 18.1\ \text{N} \quad \text{in the } -\hat{\mathbf{x}} \text{ direction}.$
• Physical interpretation:
  • The magnitude of the net force during brief interactions can be much larger than the object’s weight $mg$ (large forces acting over short times).
  • In real life, the net force arises from multiple simultaneous interactions (e.g., foot–ball contact and contact with the ground); the momentum principle only gives the total interaction, not how it splits among individual contact forces.
• Important nuance: the reported $\mathbf{F}_{\text{net}}$ is the average over $\Delta t$; the actual $F(t)$ may vary during the brief interval. A better measurement strategy could resolve this time variation.

Building a simple computer model from the estimates

• Starter code: labonestart.py (used previously for constant-velocity motion).
• Key modeling idea: represent motion in three regions with a piecewise Newton’s second law implementation:
  1) Region 1: constant velocity, $\mathbf{F}{\text{net}} = 0$. 2) Region 2 (the brief interaction): apply the average nonzero net force computed earlier; integrate over the brief interval. 3) Region 3: constant velocity again, $\mathbf{F}{\text{net}} = 0$.
• Initial conditions and parameters set at the start of the simulation:
  • Initial position and velocity for the ball, and the mass $m$.
  • Start time $T = 0$.
  • Time step for the model, $\Delta t_{ ext{model}} = 0.001\ \text{s}$ (or sometimes a larger but still small value like $0.1\ \text{s}$—the transcript notes a choice for numerical stability and resolution).
  • Note: this $\Delta t$ is used for the simulation calculations and need not equal the frame interval of the video data.
• Description of each loop in the code:
  • Loop 1: constant velocity motion with $\mathbf{F}_{\text{net}} = 0$; integrate up to the time just before the turning point (the peak).
  • Loop 2: brief interaction with $\mathbf{F}_{\text{net}}$ equal to the average force estimated from data; initial conditions come from the end of Loop 1; integrate through the interval that includes the peak (up to just after the peak’s time point).
  • Loop 3: constant velocity motion again with $\mathbf{F}_{\text{net}} = 0$ until the final observation time; use end-of-Loop 2 values as initial conditions.
• Purpose of the three-region model: to reproduce the observed qualitative behavior (ball moves in +X, turns around, then moves in -X) and to compare the model output to the observed trajectory.

Model–data comparison and interpretation

• Visualization: plot model output vs observed data; blue squares for observations, solid black line for the model.
• Overall fit: good qualitative agreement; the model captures the turn and subsequent motion.
• Quantitative discrepancy: after the turning point, the model tends to overshoot (over-predict the displacement immediately after the peak).
• Diagnostic insight:
  • A close look at the turning point shows the soccer player’s foot appears to maintain contact across multiple frames after the peak, suggesting the actual nonzero-force interval might extend beyond the assumed 1/15 s window.
  • This indicates that the simple assumption of a single constant-average $\mathbf{F}{\text{net}}$ over a fixed $\Delta t$ could be revised by adjusting the contact time or using a more detailed force profile $\mathbf{F}{\text{net}}(t)$.
• Experimental suggestion: explore different choices for the averaging interval of the net force (i.e., select different $\Delta t$) and assess whether model predictions improve.
• Broader takeaway: the exercise illustrates how to combine a fundamental principle with data analysis and simple numerical modeling to understand a brief but significant interaction.

Key formulas and numerical references (summary)

• Momentum and velocity:
  $\mathbf{p} = m \mathbf{v}.$
• Change in momentum (impulse):
  $\Delta \mathbf{p} = \mathbf{P}<em>{!f} - \mathbf{P}</em>{!i}.$
• Impulse–momentum relation (constant force assumption over $\Delta t$):
  $\Delta \mathbf{p} = \mathbf{F}_{\text{net}}\, \Delta t.$
• Net force estimate:
  $\mathbf{F}_{\text{net}} = \frac{\Delta \mathbf{p}}{\Delta t}.$
• Time intervals discussed:
  • Frame-to-frame interval:
    $\Delta t_{\text{frame}} = \frac{1}{30}\ \text{s}.$
  • Interaction interval (as estimated from the video):
    $\Delta t = \frac{1}{15}\ \text{s}.$
• Momentum values from points:
  • Initial momentum:
    $\mathbf{P}<em>{!i} = m \mathbf{V}</em>{!i},\quad \mathbf{V}<em>{!i} \approx \frac{x</em>1 - x<em>0}{t</em>1 - t_0}.$
  • Final momentum:
    $\mathbf{P}<em>{!f} = m \mathbf{V}</em>{!f},\quad \mathbf{V}<em>{!f} \approx \frac{x</em>3 - x<em>2}{t</em>3 - t_2}.$
• Example numeric result (from the transcript):
  $\mathbf{F}_{\text{net}} = 18.1\ \text{N},\quad \text{direction: } -\hat{\mathbf{x}}.$
• Three-region modeling concept:
  • Region 1: $\mathbf{F}_{\text{net}} = 0$;
  • Region 2: $\mathbf{F}_{\text{net}} = \text{(average force from data)}$;
  • Region 3: $\mathbf{F}_{\text{net}} = 0$.
• Model integration time step (example):
  $\Delta t_{\text{model}} = 0.001\ \text{s} \quad \text{or} \quad 0.1\ \text{s}.$
• Coordinate convention: initial motion along +X, turning to -X after the interaction.

Connections to foundational ideas and real-world relevance

• Builds directly on Newton’s second law expressed in momentum form (impulse–momentum theorem).
• Demonstrates a practical method for analyzing brief interactions in sports physics and other collision-like events.
• Illustrates how data analysis (position vs time) informs momentum estimates and how those in turn constrain forces via the impulse relation.
• Highlights the interplay between measurement precision, modeling assumptions (constant $\mathbf{F}_{\text{net}}$ over a short interval), and the interpretation of results.
• Real-world relevance: understanding ball–foot–ground interactions can inform coaching, equipment design, and safety analyses in sports.

Practical and ethical/philosophical implications

• Measurement uncertainty matters: the exact start/end of the interaction is inferred from frames; faster measurements could change estimated $\Delta t$ and $\mathbf{F}_{\text{net}}$.
• Modeling simplifications (e.g., constant $\mathbf{F}_{\text{net}}$ during $\Delta t$) are useful for tractable predictions but may miss time-varying force profiles; one should test robustness by varying assumptions.
• Cumulative models vs. decomposing forces: the approach emphasizes that physics can describe the net effect without necessarily identifying all microscopic interactions; this reflects a practical stance in systems where detailed force channels are complex or inaccessible.
• Encourages iterative refinement: compare model predictions with data, revise the assumed interaction window, or adopt more sophisticated force profiles to improve fidelity.

Connections to previous lectures and broader context

• Reuses the impulse–momentum framework introduced earlier, extending it to a concrete brief collision scenario.
• Connects kinematics (position–time data) with dynamics (forces and momentum) through the intermediate step of estimating velocities and momenta from measurements.
• Demonstrates a standard modeling pattern: identify regimes (constant velocity, brief impulse, then constant velocity), solve piecewise, and compare with data to judge adequacy.