Module 2: Kinematics, Units, and Tracking in Sports

Module overview and lab focus

The week introduces Module 2 with a shift from foundational modeling to kinematics (distance, displacement, velocity, acceleration) and the units that underpin measurements. Labs this week centre on camera-based performance capture using vertex devices to measure a counter-movement jump. Students will define a performance objective for the vertex, label events, phases, and critical elements of the movement, and work in small groups. Early lab tasks include recording a counter-movement jump, producing accurate video, using the manual zoom function on cameras, and performing performance corrections after several trials. The aim is to build a robust performance model, identify errors, and learn to prioritise the order of corrections discussed previously. In weeks 1–3 (Module 1) the focus was on building the model and diagnosing errors in lecture content; Module 2 (this week) introduces more heavy concepts: displacement, velocity, and acceleration, along with extensive use of interactive tools and videos. A key theme is understanding kinematics (movement in terms of length and time) and kinetics (forces associated with movement) as two related but distinct domains.

From today we also introduce units and the standard measures used to quantify motion. We’ll explore velocity and acceleration as changes in displacement and time, and discuss how new motion-tracking technologies (including 3D motion capture) provide deeper insight into velocity and acceleration. There will be emphasis on the distinction between scalar and vector quantities, the difference between kinematics and kinetics, and the relationship between movement and the forces that produce it. The course will also touch on real-world applications using Usain Bolt’s performances as a review example to illustrate how velocity and acceleration vary across a sprint and what that means for performance. The module foregrounds that the basis of kinematics is length and time, and later modules will cover kinetics and forces with devices like force plates and 3D motion capture.

Kinematics vs kinetics; what we study when

Kinematics deals with movement without regard to the forces that cause it. The central quantities are displacement, velocity, and acceleration. Displacement is a vector: it has magnitude and direction, representing the change in position from start to end. Velocity is the rate of change of displacement with time and is also a vector. Acceleration is the rate of change of velocity with time and is a vector.

Kinetics, by contrast, focuses on the forces that produce movement. Mass and forces (including gravity, friction, air resistance, and the coefficient of restitution in collisions) are central topics. The course emphasizes external forces (gravity, friction, air resistance) rather than internal forces in this module.

Key relationships to hold in mind:

Distance is a scalar quantity: how far you move in total, regardless of direction.
Displacement is a vector: the straight-line change in position from start to end.
Velocity is a vector: the rate and direction of motion.
Speed is a scalar: the magnitude of velocity (how fast, without direction).
Acceleration is a vector: the rate of change of velocity.

In later weeks, kinetics will introduce the master equation of forces and how mass interacts with distance and time. For now, the course focuses on how kinematics and measured quantities relate to the external forces experienced by a moving body.

Units and measurements; the language of motion

We use standard metric units across biomechanics and physics. Distance and displacement are measured in meters (m). Time is measured in seconds (s). Velocity has units of meters per second ( ext{m s}^{-1} ) and acceleration has units of meters per second squared ( ext{m s}^{-2} ). In the lecturer’s notation, velocity is abbreviated as ms⁻¹ (i.e., 1 ext{ ms}^{-1}) and acceleration as ms⁻². In this course, the formulae are not memorised in isolation; a formula sheet is provided and used in exams to apply the correct relation.

Examples of unit usage include:

Velocity: v = rac{ ext{change in displacement}}{ ext{change in time}} = rac{Δs}{Δt} with units rac{ ext{m}}{ ext{s}}.
Acceleration: a = rac{Δv}{Δt} with units rac{ ext{m}}{ ext{s}^2}.

A key practical point discussed is that exams include a dedicated section on units (about 30 marks available for calculations, with 30 unit marks noted in the lecture). Students are reminded not to memorize every formula but to know how to select and apply the right formula from the provided sheet. The instructor emphasises the distinction between metric and imperial units and explains that the course uses metric units as the standard.

Kinematics components: distance, displacement, velocity, and acceleration

Distance is the total length traveled, regardless of direction. Displacement is the straight-line vector from the starting point to the endpoint and depends on direction. For example, if you run 100 m on a track, your distance is 100 m; if you started in the blocks of a 100 m dash and ended at the same line after completing the loop, your displacement could be zero because you ended at your starting position.

Velocity measures how fast and in what direction you move between two points, and is calculated as the change in displacement over time, with units of ext{m s}^{-1}. Acceleration measures how quickly velocity changes, with units of ext{m s}^{-2}. The lecturer explains the sign conventions: positive displacement/velocity indicates movement away from the origin in the chosen coordinate system, while negative displacement/velocity indicates movement back toward the origin. The slope (gradients) on displacement-time graphs indicates velocity: a steep gradient means high velocity, whereas a shallow gradient indicates slower movement. Acceleration shows up as the curvature or changes in the velocity graph; a rapid change in velocity over a short time corresponds to high acceleration.

Student-friendly summaries:

Displacement is the vector from start to finish; distance is total path length.
Velocity is the rate of change of displacement; speed is the magnitude of velocity.
Acceleration is the rate of change of velocity.

The session also clarifies that graphs of motion are analyzed by looking at gradients: the gradient of a displacement-time graph equals velocity, while the gradient of a velocity-time graph equals acceleration. A curved area on a displacement-time graph signifies increasing velocity (positive acceleration), while a straight line with zero slope indicates zero velocity (no net movement over time). In a velocity-time graph, a horizontal line (zero slope) indicates constant velocity (zero acceleration).

Scalars vs vectors; momentum and coordinate systems

Scalars (like distance, mass, time, area) describe magnitude only. Vectors (like displacement, velocity, acceleration, momentum) describe magnitude and direction. Momentum, in particular, is a vector defined as oldsymbol{p} = moldsymbol{v}, where velocity carries direction and magnitude and mass contributes to the momentum of a moving object. Momentum is especially relevant in collisions and sports actions such as rugby tackles, ball throws, and impacts in cricket and tennis.

The coordinate system used in the Canovia example places the origin at (0,0) with the top-right quadrant representing positive x and positive y directions. The system is used to quantify movement on a plane: distance along the ground (x) and height or vertical displacement (y). The lecturer demonstrates how signs change as movement passes through the origin, explaining that the positive quadrant is the usual working area in this course’s Canovia environment.

Tracking technologies: GPS and 3D athlete tracking

GPS has evolved from large, carry-sized systems used in early military contexts (e.g., Desert Storm) to lightweight trackers embedded in watches, phones, and other wearables. A minimum of four satellites is typically required to pinpoint a position accurately. GPS provides distance and time, from which velocity and acceleration can be inferred, enabling assessment of movement patterns, high-intensity running, and loads (player load, fatigue) across quarters in team sports. The lecture gives a historical nod to the Gulf War where GPS markers were crucial for tracking in sandstorms, illustrating the robustness of GPS in challenging environments. Modern GPS-based systems classify movement into speed zones (colors such as red for high speed, yellow for moderate, blue/green for lower speeds) and are often attached near the base of the neck (the C7 vertebra) to minimize movement artifacts.

The session also introduces 3D motion capture and 3D athlete tracking, with demonstrations of AI-based skeleton extraction from two-camera setups to visualize joint movements and forces throughout an event. The use of 3D tracking allows visualizing how movement interacts with forces and how force plates translate body mass into distance and time measurements. These tools enable richer visual representations for understanding kinematics and kinetics in sports.

Displacement, velocity, and acceleration in practice; Usain Bolt as a case study

Throughout the Bolt case discussions, the class examines how displacement and velocity evolve during a sprint. Standing at the starting line, the gradient is steep in the early metres, indicating rapid acceleration, followed by a curved ascent as velocity increases towards top speed (roughly around 60 m for many sprinters in a race). The Bolt analyses show that once top speed is reached, maintaining it matters; a late peak in velocity and a slower end-phase can influence overall time. In the 2008 Olympic race Bolt reached top speed midway and then decelerated at the end, but still won due to his earlier acceleration advantages. A later 2009 race showed a different pattern: Bolt maintained higher velocity for longer and did not slow as much at the end, achieving a faster final time than in 2008.

The instructor notes that there are published mathematical models predicting what Bolt’s time could have been if he had maintained top speed longer or altered acceleration profiles. A common estimate around the time of the lecture was a potential time near 9.6 s for the 100 m if top speed were sustained longer; actual events show top speeds and accelerations that shape outcomes and times.

The Bolt discussions integrate several measurement tools: timing gates (with 10 m splits up to 100 m and beyond), velocity and acceleration curves, and the interpretation of how early acceleration translates into end-of-race performance. In addition to the sprint analysis, the discussion ties in with tracking technologies, including 3D athlete tracking videos shown in class, which visualize speed zones and joint movements.

Trigonometry in projection and decomposition of velocity

A key practical application addressed is decomposing a velocity vector into horizontal and vertical components for projectile-like motions (e.g., basketball free throw, javelin, long jump, hammer throw). The general approach is to use trigonometry to resolve a release velocity into its components:

Horizontal component: v_x = v \, ext{cos}\theta
Vertical component: v_y = v \, ext{sin}\theta

The instructor emphasises two crucial points for exams:

Use sine for the vertical component and cosine for the horizontal component when the angle of release is known, not tangent. Tangent involves the ratio of vertical to horizontal components and can introduce errors if the sides are not known in advance.
In exams, you should be comfortable with the signs and the relative magnitudes: if the horizontal component is larger than the vertical, the resultant motion is more horizontal; if vertical is larger, it reflects a higher launch angle.

The classroom also discusses the broader physics interpretation: a basketball free throw requires more vertical component to reach the basket, whereas a long pass or a drive in a golf shot emphasizes horizontal carry. Spin can also modify trajectory and landing behavior (backspin, for example, can influence the way a ball interacts with a surface). These examples illustrate how the decomposition of velocity into components informs strategy and technique in sports.

A practical caution is given about calculators: in exams you should rely on sine and cosine buttons for the vertical and horizontal components, respectively. The tangent function should be avoided for resolving components since it can lead to compounding mistakes if you misidentify the opposite or adjacent sides. The lecturer warns about calculator reliability and recommends familiarising yourself with the basic trigonometric functions and predetermining which button to press for each component.

Graphical interpretation of motion; a moving-man demonstration and beyond

A moving-man simulation is presented as a tool to explore relationships between position, velocity, and acceleration. The activity demonstrates how different movement patterns translate into different shapes on a position-time graph and how velocity and acceleration correspond to the slope and curvature of those graphs. Observations include:

A steeper gradient in the position-time graph corresponds to a higher velocity.
A curved segment in the position-time graph indicates changing velocity (acceleration).
The velocity-time graph shows where acceleration is positive, negative, or zero (zero acceleration when velocity is constant).

The lecture complements this with a real-time YouTube-style animation to reinforce the concept that the gradient on a displacement-time graph reveals how quickly the position changes and thereby indicates velocity. The presenter also discusses an example of a city route with varying speeds and traffic signals to illustrate positive and negative acceleration scenarios as one proceeds along a path.

In addition to the moving-man tool, the session introduces online resources such as interactive simulations (e.g., projectile motion simulations and rate-of-change tools) to let students observe how position, velocity, and acceleration evolve in a controlled environment.

Practical implications, ethics, and real-world relevance

The work relies on wearable tracking and imaging technologies (GPS, 3D motion capture) that collect sensitive data about athletes’ performance. While offering rich performance insights, it also carries practical considerations about privacy, data ownership, and the appropriate use of performance data.
In exams and labs, students are reminded not to rely on memorization but to use the provided formula sheet and to practice applying formulas to real-world data (e.g., Usain Bolt race data, split times from timing gates).
The story of Usain Bolt, Usain Bolt’s performances in 2008 and 2009 and subsequent modeling work, highlights the importance of modeling and prediction in sports science, where data-driven estimates can guide training decisions and performance expectations.
The discussion touches on how different sports demand different velocity components (horizontal vs vertical) depending on the task (distance travel vs vertical elevation) and how this informs technique and equipment choice (e.g., club loft, ball spin).
Finally, the course emphasizes the practicalities of data collection in labs: camera use, video quality, and corrections, all designed to ensure reliable measurement and meaningful interpretation.

Summary of key formulas and concepts (LaTeX)

Velocity as a rate of displacement:
oldsymbol{v} = rac{Δoldsymbol{s}}{Δt}
Acceleration as a rate of velocity change:
oldsymbol{a} = rac{Δoldsymbol{v}}{Δt}
Components of velocity (release velocity decomposed):
vx = v \, ext{cos}\theta,\n\quad vy = v \, ext{sin}\theta
Magnitude of a resultant vector from components: if components are $vx, vy$, then
|oldsymbol{v}| = \, ext{√}(vx^2 + vy^2)
Displacement as a vector between start and end:
Δoldsymbol{r} = oldsymbol{r}2 - oldsymbol{r}1
Momentum (vector):
oldsymbol{p} = moldsymbol{v}
Distance vs displacement (conceptual): distance is path length; displacement is the straight-line change in position.
Coordinate sign conventions: in the Canovia example, the origin is at (0,0); the top-right quadrant is positive for both x and y, representing movement away from the origin in both directions.

Connections to prior and real-world concepts

The module ties into overarching physics principles: motion arises from the combination of position, velocity, and acceleration; kinetics involves the forces producing those motions—gravity, friction, air resistance, and collision responses.
Real-world relevance is seen in sport science, where tracking technologies quantify movement (e.g., sprint analysis, ball trajectory, jump height), and in rehabilitation or performance optimization where data-driven feedback informs training regimens.
The integration of graphs, coordinate systems, and vector decomposition mirrors standard physics problem-solving strategies and supports the translation of classroom concepts into on-field decision-making (e.g., choosing a technique based on horizontal vs vertical velocity components).

Practical lab and assessment notes

Labs this week require students to prepare a performance model for the vertex device, collect high-quality video, and perform iterative corrections to improve measurement accuracy.
Exams will provide a formula sheet; memorization isn’t required, but understanding when and how to apply formulas is essential.
Students are encouraged to explore a suite of interactive tools and video resources to visualise concepts such as velocity, acceleration, and the impact of different projection angles in sport.
The course also notes differences between linear and angular motion; angular motion involves rotation about a fixed axis and will be addressed in later modules.

Quick reference: glossary of terms (concepts you should know)

Displacement: a vector describing the straight-line change in position from start to end.
Distance: the total length traveled along a path (scalar).
Velocity: rate of change of displacement; a vector with magnitude and direction.
Speed: magnitude of velocity (scalar).
Acceleration: rate of change of velocity; a vector.
Scalar: quantity with magnitude only (e.g., distance, mass, time).
Vector: quantity with magnitude and direction (e.g., displacement, velocity, acceleration, momentum).
Coordinate system: a framework (origin and axes) used to locate movement in space.
Components: projection of a vector onto axes (e.g., horizontal and vertical components).
Projection angle: the angle of release or launch that determines the split between vertical and horizontal components.
Pythagoras and trigonometry: used to resolve vectors into components and compute resultant magnitudes when necessary.
GPS and 3D tracking: technologies for measuring distance, time, velocity, and acceleration in real-world sport settings.

Lab-ready takeaways

Expect to define and label M1-style performance models for cameras, with a focus on objective setting and error correction priorities.
Expect to work with measurement units; be ready to derive components and magnitudes from given data using the sine/cosine approach rather than relying on tangent unless the sides are known.
Be prepared to explain the difference between displacement and distance with concrete examples (e.g., track vs pool scenarios).
Be able to discuss how slope/gradient on displacement-time graphs translates to velocity, and how velocity-time graphs relate to acceleration.

These notes reflect the breadth of the lecture content: an introduction to Module 2 that links practical lab tasks to core kinematic concepts, emphasizes the correct use of units and formulas, and connects theory to real-world sport applications through examples like Usain Bolt and projection-based sports activities.