UPI · First-year undergraduate · Calculus-based mechanics
Mechanics with calculus, evidence, and physical meaning.
A rigorous first course in mechanics for physics, engineering, chemistry, mathematics, and related STEM students. It develops physical modelling from measurement and vectors through Newtonian mechanics, energy, momentum, rotation, gravitation, and oscillations. Fluids remains a numbered but institution-dependent option, with a separate unnumbered Waves & Sound bridge where local syllabi require it.
Independent pathway notice. GioPhysics University & College Physics is an independent learning pathway. GioPhysics is not a university, does not award academic credit or degrees, and is not accredited by or affiliated with any university or professional body. This is not a universal programme specification. Course coverage and degree requirements vary by institution; confirm requirements with your university.
Before unit 01
Make the mathematical starting point visible.
University Physics I is calculus based. A learner does not need every technique to feel automatic on day one, but the ideas below should be familiar enough to practise alongside the physics.Readiness, not gatekeeping
Know what to refresh before it becomes a barrier.
If several items are unfamiliar, begin with a mathematics bridge or introductory mechanics course. If they are rusty, revisit them just in time as each unit calls for them.
- Recommended: algebra, trigonometry, vector components, functions, and scientific notation
- Single-variable calculus completed or taken alongside the course
- No multivariable calculus is assumed
- Comfort interpreting tables, graphs, and equations
How calculus enters: derivatives describe instantaneous rates and local change; integrals accumulate change and extend work, impulse, centre-of-mass, and rotational models. The course should connect each operation to a graph, units, and a physical interpretation—not only a memorized rule.
Mathematical tools used explicitly
- Vector components, unit vectors, dot and cross products
- Derivatives as rates of change
- Definite integrals as accumulated change
- Elementary differential equations for constant-acceleration and oscillation models
- Dimensional, limiting-case, and order-of-magnitude checks
Five connected phases
Build the model before raising the difficulty.
The sequence moves from representation and kinematics to interactions, conservation, rotation, and continuous or periodic systems.Units 1–11 form the recommended mechanics core, from mathematical foundations through angular momentum.
Units 12–14—static equilibrium, gravitation, and oscillations—are commonly included, but their order and depth vary by institution.
Unit 15, Fluids, is an institution-dependent option and may instead appear in another physics or engineering course.
Course scope: This GioPhysics course map is an adaptable teaching sequence, not a claim of accreditation or a universal university syllabus. Departments can adjust the order, mathematical depth, laboratory hours, and optional fluids endpoint to match local requirements.
Complete numbered course map + unnumbered extension
All fifteen units, searchable in a reasoned order.
Search 120 numbered subsections, inspect every outcome and laboratory direction, and open the separate 11-topic Waves & Sound extension only when it belongs in your institution's endpoint.Mapped lesson links: linked topics open relevant existing GioPhysics material, which may span several secondary and advanced lesson collections. Use those lessons as conceptual or problem-solving support while continuing to follow your institution's depth, notation, laboratory, and assessment requirements.
Interactive course map
Choose one phase. Open one unit. Learn by doing.
Search includes unit titles, topics, skills, outcomes, prerequisites, and laboratory questions across all 15 units and the unnumbered institutional extension.
Private study checklist
0 of 15 units reviewedUPI-01 · Foundation
Mathematical & Physical Foundations
Establish SI measurement, dimensional reasoning, uncertainty, vectors, coordinate systems, calculus, and estimation as the working language of the course.
9 topics6 outcomes2 lab directions100 min practiceOpen Mathematical & Physical Foundations: topics, evidence, and practice
Topics and mapped lessons
- SI quantities, units, and prefixes
Base and derived quantities, prefix conversions, coherent units, and unambiguous numerical reporting.
- Dimensional analysis and scaling
Dimensional consistency, simple power-law models, proportional reasoning, and scale estimates.
- Significant figures and measurement uncertainty
Resolution, repeatability, accuracy and precision, random and systematic effects, uncertainty intervals, propagation in derived quantities, and appropriate reporting.
- Coordinate systems and components
Origin and axis choices, Cartesian components, signs, and changing the description without changing the physics.
- Vectors and unit-vector notation
Magnitude, direction, vector addition, unit vectors, and graphical versus component methods.
- Dot and cross products
Geometric and component definitions, physical interpretation, and preparation for work and torque.
- Derivatives and integrals in physical models
Rates, accumulated change, initial conditions, graphical meaning, and numerical approximation.
- Estimation and model checks
Order-of-magnitude reasoning, limiting cases, unit checks, and physical plausibility.
- Data analysis and model comparison
Tables and graphs, fitted parameters, residuals, uncertainty, and evidence-based comparison of competing physical models.
Learning outcomes
- Convert measurements and derived quantities consistently within SI.
- Use dimensions to test equations and infer the form of simple relationships.
- Resolve vectors and express them with components or unit-vector notation.
- Interpret a derivative as an instantaneous rate and an integral as accumulated change.
- Report a result with sensible significant figures, uncertainty, and units.
- Use plots, fitted parameters, residuals, and uncertainty to compare simple physical models.
Prerequisite thread
- High-school algebra and trigonometry
- Scientific notation and graph reading
Laboratory directions
hands-onMeasure what cannot be read directly Which combination of direct measurements gives the most defensible value for an irregular object's density?
Evidence: Repeated measurements, uncertainty bounds, and a comparison of two methodscomputationalDerivative and area from motion data How accurately can numerical slopes and areas reconstruct a known motion?
Evidence: Sampled data, derivative and integral estimates, residuals, and a step-size discussionProblem practice
Units, estimates, vectors, and calculus meaning before formula substitution
Capstone: Estimate a measurable campus quantity using two independent models and reconcile the results.
- Concept checks that require an explanation, not only a numerical answer
- Multi-step quantitative modelling with unit and limiting-case checks
- A calculus or data-interpretation challenge
Interactive concept map
Follow the model from claim to evidence.
01 · Physical claimMathematical & Physical Foundations Establish SI measurement, dimensional reasoning, uncertainty, vectors, coordinate systems, calculus, and estimation as the working language of the course.
02 · RepresentationScale sketches Dimensional analysis
03 · TestPrediction before measurement Which combination of direct measurements gives the most defensible value for an irregular object's density?
04 · Evidence & boundaryDecide, then qualify Repeated measurements, uncertainty bounds, and a comparison of two methods
Interactive diagram for Mathematical & Physical Foundations: follow the physical claim through its representation, proposed test, evidence, and model boundary. modelDimensional analysis turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
Concept Using Mathematical & Physical Foundations, explain how a physicist can: Convert measurements and derived quantities consistently within SI.
- SI quantities, units, and prefixes
UPI-02 · Heavy
1D Kinematics
Describe one-dimensional motion using position, velocity, acceleration, motion graphs, constant-acceleration models, free fall, and calculus.
8 topics4 outcomes2 lab directions140 min practiceOpen 1D Kinematics: topics, evidence, and practice
Topics and mapped lessons
- Position, displacement, and distance
Reference frames, origins, path length, displacement, and careful interval notation.
- Average and instantaneous velocity
Secant and tangent slopes, direction, speed, and the limiting idea behind instantaneous velocity.
- Average and instantaneous acceleration
Velocity change per time, signed acceleration, and why negative acceleration is not automatically slowing down.
- Position, velocity, and acceleration graphs
Slope and area relationships, turning points, discontinuities, and translating among graphs.
- Constant-acceleration equations
Derivation and use of the standard relations, with initial conditions and applicability stated.
- Free fall and vertical motion
A near-Earth constant-g model, upward and downward motion, release versus launch, and limitations.
- Kinematics with calculus
Deriving velocity and acceleration, integrating with initial conditions, and treating nonconstant acceleration.
- Piecewise and inverse motion problems
Multiple intervals, continuity conditions, unknown graphs, and reconstructing motion from constraints.
Learning outcomes
- Distinguish position, displacement, distance, average velocity, and average speed.
- Relate position, velocity, and acceleration using graphs and calculus.
- Solve constant-acceleration and free-fall problems with a declared positive direction.
- Construct a piecewise motion model and test it against boundary conditions.
Prerequisite thread
- Signed quantities and coordinate choices
- Slopes, areas, derivatives, and simple integrals
Laboratory directions
video analysisCart motion from video When is a cart's acceleration adequately constant?
Evidence: Position-time data, velocity estimates, model fit, residuals, and scale uncertaintyhands-onFree fall without assuming the answer What value of gravitational acceleration is supported by the data?
Evidence: Multiple trials, a linearized graph, best-fit slope, and uncertainty comparisonProblem practice
Selecting a representation and connecting graph features to equations
Capstone: Reconstruct an object's complete motion from incomplete sensor graphs and justify every interval.
- Concept checks that require an explanation, not only a numerical answer
- Multi-step quantitative modelling with unit and limiting-case checks
- A calculus or data-interpretation challenge
Interactive concept map
Follow the model from claim to evidence.
01 · Physical claim1D Kinematics Describe one-dimensional motion using position, velocity, acceleration, motion graphs, constant-acceleration models, free fall, and calculus.
02 · RepresentationMotion diagrams Motion diagrams
03 · TestPrediction before measurement When is a cart's acceleration adequately constant?
04 · Evidence & boundaryDecide, then qualify Position-time data, velocity estimates, model fit, residuals, and scale uncertainty
Interactive diagram for 1D Kinematics: follow the physical claim through its representation, proposed test, evidence, and model boundary. modelMotion diagrams turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
Concept Using 1D Kinematics, explain how a physicist can: Distinguish position, displacement, distance, average velocity, and average speed.
- Position, displacement, and distance
UPI-03 · Heavy
2D & 3D Kinematics
Extend motion to vector position, velocity, and acceleration; treat projectiles, relative velocity, and circular motion using component models.
8 topics4 outcomes2 lab directions150 min practiceOpen 2D & 3D Kinematics: topics, evidence, and practice
Topics and mapped lessons
- Vector position, velocity, and acceleration
Vector-valued functions, component derivatives, paths, and vector directions.
- Projectile motion
Component models, launch geometry, trajectory equations, and complementary-angle results.
- Relative position and velocity
Frame labels, vector subtraction, moving media, pursuit, and closest approach.
- Uniform circular motion
Angular speed, period, tangential velocity, centripetal acceleration, and vector direction.
- Tangential and radial acceleration
Changing speed and direction in curved motion using local components.
- Motion in three dimensions
Cartesian vector functions, space curves, relative motion, and coordinate choice.
- Kinematic constraints
Connected motions, geometry-based constraints, and differentiating constraint relations.
- Multidimensional motion problem studio
Unknown launch conditions, relative-motion intersections, circular arcs, and model checks.
Learning outcomes
- Differentiate and integrate vector-valued motion functions.
- Model projectile motion by separating horizontal and vertical components.
- Transform velocities between translating reference frames.
- Describe circular motion with tangential and radial acceleration components.
Prerequisite thread
- Vector components and unit vectors
- One-dimensional calculus-based kinematics
Laboratory directions
video analysisProjectile model versus trajectory Over what range does the no-drag projectile model describe a launched ball?
Evidence: Tracked x and y data, separate fits, residuals, and a model boundarycomputationalRelative-motion navigation challenge Which heading reaches a target fastest in uniform wind or current?
Evidence: A vector model, parameter sweep, trajectory plot, and analytic comparisonProblem practice
Decomposing vector motion without treating components as separate events
Capstone: Design a launch or navigation plan that meets two constraints and quantify its sensitivity to one assumption.
- Concept checks that require an explanation, not only a numerical answer
- Multi-step quantitative modelling with unit and limiting-case checks
- A calculus or data-interpretation challenge
Interactive concept map
Follow the model from claim to evidence.
01 · Physical claim2D & 3D Kinematics Extend motion to vector position, velocity, and acceleration; treat projectiles, relative velocity, and circular motion using component models.
02 · RepresentationTrajectory sketches Vector-valued functions
03 · TestPrediction before measurement Over what range does the no-drag projectile model describe a launched ball?
04 · Evidence & boundaryDecide, then qualify Tracked x and y data, separate fits, residuals, and a model boundary
Interactive diagram for 2D & 3D Kinematics: follow the physical claim through its representation, proposed test, evidence, and model boundary. modelVector-valued functions turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
Concept Using 2D & 3D Kinematics, explain how a physicist can: Differentiate and integrate vector-valued motion functions.
- Vector position, velocity, and acceleration
Labs and problem solving
Use one repeatable loop from page to physical system.
Laboratory work and analytical problems are different forms of the same discipline: choose a model, make assumptions explicit, test the result, and interpret what survives.Represent
Sketch the system, choose coordinates, define symbols, and identify the model boundary.
Derive
Connect definitions and laws symbolically before substituting numerical values.
Solve
Use algebra, vectors, derivatives, integrals, approximations, or computation deliberately.
Test
Check units, signs, scale, limiting behaviour, and agreement with experimental evidence.
Interpret
Explain what the result says physically, where the model works, and where it can fail.
Course-level outcomes
What successful study should make possible.
Outcomes describe transferable capabilities rather than promising a university grade, credit award, or professional qualification.- Translate a physical situation into a system boundary, assumptions, diagram, and mathematical model.
- Use derivatives and integrals to connect position, velocity, acceleration, force, work, impulse, and angular quantities.
- Apply Newton's laws, conservation laws, and rotational dynamics to multi-object systems.
- Move fluently between words, diagrams, graphs, equations, numerical results, and physical interpretation.
- Plan and evaluate measurements using uncertainty, residuals, and model limitations.
- Use computational or graphical methods when an analytic solution is unavailable or inefficient.
- Check units, signs, magnitudes, limiting cases, and conservation constraints before accepting a result.
- Communicate a defensible solution or laboratory conclusion with assumptions and uncertainty made visible.
Choose the right starting point
Ready for the mathematics? Begin at unit 01.
Start with measurement and uncertainty even if the notation looks familiar. It establishes the standards used to judge models, graphs, laboratory evidence, and numerical results in every later unit.