Dynamics: Kinematics - Concept Map
Kinematics: Description of Motion Without Forces
Related concepts
Rectilinear Motion
Sub concepts
- Constant Acceleration Equations
- Variable Acceleration Integration
- Velocity Definition (ds/dt)
- Acceleration Definition (dv/dt)
- Free Fall (g = 9.81 m/s²)
Relationship to central
Core kinematic form; motion along a straight line with constant or variable acceleration
Projectile Motion
Sub concepts
- Horizontal Motion (Constant Velocity)
- Vertical Motion (Constant Acceleration)
- Range Calculation
- Maximum Height
- Time of Flight
- Launch Angle Effects
- Independence of Motion Components
Relationship to central
Two-dimensional kinematic application; combines horizontal and vertical rectilinear motions
Rotational (Angular) Kinematics
Sub concepts
- Angular Displacement (θ, radians)
- Angular Velocity (ω, rad/s)
- Angular Acceleration (α, rad/s²)
- Constant Angular Acceleration
- Linear-Angular Relationships (v = rω, a_t = rα)
- Centripetal Acceleration (a_n = v²/r = rω²)
- Revolution-to-Radian Conversion
Relationship to central
Kinematics applied to rotating bodies; uses angular analogs of rectilinear parameters
Fundamental Kinematic Parameters
Sub concepts
- Position (s, meters)
- Velocity (v, m/s)
- Acceleration (a, m/s²)
- Time (t, seconds)
- Initial Conditions (v₀, s₀)
Relationship to central
Building blocks of all kinematic descriptions
Mathematical Relationships
Sub concepts
- Differentiation (v = ds/dt, a = dv/dt)
- Integration (v = ∫a dt, s = ∫v dt)
- Constant-a Kinematic Trio
- Elimination of Time Variable
- Trigonometric Applications (Projectile Motion)
Relationship to central
Quantitative tools that define kinematic equations
Common Board-Exam Problem Types
Sub concepts
- Vehicle Acceleration/Braking
- Free-Fall Height/Impact Speed
- Projectile Clearance (Obstacles)
- Rotating Equipment (Flywheels, Wheels)
- Multi-Stage Motion Sequences
Relationship to central
Practical assessment contexts for kinematic analysis
Concept connections
- Rectilinear MotionFundamental Kinematic ParametersPosition, velocity, and acceleration are core parameters that rectilinear motion describes through time-dependent functions.
- Rectilinear MotionFree FallFree fall is a special case of rectilinear motion where acceleration is constant at g = 9.81 m/s² downward.
- Projectile MotionRectilinear MotionProjectile motion decomposes into two independent rectilinear motions: constant horizontal velocity and constant vertical acceleration (g).
- Projectile MotionTrigonometric ApplicationsLaunch angle θ determines initial velocity components (v₀cosθ horizontal, v₀sinθ vertical) and appears in range and height formulas via sin2θ and sin²θ.
- Rotational KinematicsFundamental Kinematic ParametersAngular analogs (θ, ω, α) mirror rectilinear (s, v, a), providing parallel kinematic descriptions for rotating bodies.
- Rotational KinematicsLinear-Angular RelationshipsA point at radius r on a rotating body links angular and linear motion: v = rω, a_t = rα, a_n = rω².
- Mathematical RelationshipsRectilinear MotionCalculus (derivatives and integrals) establishes the definitions v = ds/dt, a = dv/dt and enables solution of variable-acceleration problems.
- Mathematical RelationshipsRotational KinematicsSame calculus relationships apply: ω = dθ/dt, α = dω/dt, enabling solution of variable angular acceleration.
- Constant Acceleration EquationsRectilinear MotionThe three kinematic equations (v = v₀ + at, s = v₀t + ½at², v² = v₀² + 2as) are the primary tools for constant-a problems.
- Constant Angular AccelerationRotational KinematicsAngular analogs of the three kinematic equations (ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ) solve constant α problems.
- Independence of Motion ComponentsProjectile MotionHorizontal and vertical motions in a projectile are independent, sharing only time t; this is fundamental to projectile analysis.
- Launch Angle EffectsRange CalculationThe range formula R = v₀²sin2θ/g shows that range depends critically on angle; maximum at 45°.
- Variable Acceleration IntegrationRectilinear MotionWhen acceleration is not constant, integration of a(t) or a(v) or a(s) replaces the three kinematic equations.
- Revolution-to-Radian ConversionRotational KinematicsConversion 1 rev = 2π rad and ω (rad/s) = 2πN/60 from rpm is essential before applying angular equations.
- Centripetal AccelerationRotational KinematicsA point on a rotating body has centripetal (normal) acceleration a_n = v²/r = rω² directed toward the axis.
- Common Board-Exam Problem TypesRectilinear MotionVehicle braking, free-fall height, and impact speed calculations are standard rectilinear kinematics problems.
- Common Board-Exam Problem TypesProjectile MotionProjectile clearance (clearing obstacles, walls) and impact angle calculations are typical board-exam applications.
- Common Board-Exam Problem TypesRotational KinematicsFlywheel acceleration, wheel braking, and rpm-to-rad/s conversions are routine rotating-equipment problems.
- Sign ConventionsRectilinear MotionConsistent choice of positive direction (up, down, left, right) determines the sign of velocity and acceleration; critical for correctness.
- Time Variable EliminationConstant Acceleration EquationsThe equation v² = v₀² + 2as eliminates time, allowing direct solution when t is not given or not useful.
Diagrams
Kinematics: Complete Concept Hierarchy and Relationships
Complete hierarchical mind map showing all major kinematic concepts, their definitions, key equations, and relationships. Central theme is that kinematics describes motion (position, velocity, acceleration) without considering forces. Three main branches (rectilinear, projectile, rotational) each decompose into fundamental parameters, governing equations, and practical applications. The base layer covers mathematical tools (calculus, sign conventions) and problem-solving strategy.
Rectilinear Motion Problem-Solving Flowchart
Decision tree for solving rectilinear motion problems. Starts by determining whether acceleration is constant. If constant, selects the appropriate kinematic equation based on which variables are known. If variable, sets up differentiation or integration. Ends with verification of units and sign consistency—critical for board exams where sign errors are common.
Projectile Motion Analysis Sequence
Step-by-step flowchart for projectile motion analysis. Emphasizes component resolution, selection of appropriate formulas based on what is requested (range, height, time, or clearance), and the critical decision of whether launch and landing heights differ. Highlights the 45° optimum and the independence of horizontal and vertical motions.
Rotational Kinematics: From rpm to Linear Motion
Detailed flowchart for rotational kinematics problems. Emphasizes critical unit conversion from rpm to rad/s (factor 2π/60). Shows selection of constant angular acceleration equations, then demonstrates conversion from angular to linear quantities using radius. Includes calculation of centripetal acceleration and the total acceleration of a point on the rotating body.
Three Kinematic Equations: When to Use Each
Decision guide for selecting the correct kinematic equation when constant acceleration is involved. Maps unknown variables to appropriate formulas. The three core equations (v = v₀ + at, s = v₀t + ½at², v² = v₀² + 2as) handle all constant-acceleration scenarios. Shows the strategy of combining equations when more than one is needed.
Motion State Transitions in Kinematics Problems
State diagram showing the flow of a kinematic analysis from initial conditions through selection of equations (constant vs. variable acceleration), calculation of intermediate states, and determination of final motion state. Demonstrates that motion can be composed of multiple phases, each with its own initial conditions and kinematic equations.
Projectile vs. Rectilinear vs. Rotational: Decision Tree
Classification flowchart distinguishing between the three major kinematic problem types: rectilinear motion (one-dimensional, non-rotating), rotational kinematics (angular motion), and projectile motion (two-dimensional with gravity). Each branch specifies the defining equations, key special cases, and typical problem contexts.
Kinematic Problem Distribution on Board Exams
Estimated distribution of kinematics problem types on the PRC Civil Engineer Licensure Examination. Rectilinear motion with constant acceleration dominates (35%), followed by projectile motion (25%), free-fall applications (20%), and rotational kinematics (15%). Variable acceleration and multi-stage problems appear rarely (5%), but understanding them deepens conceptual mastery. This distribution guides study prioritization.