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Glossary
Chapter 1: What mathematical modelling is
- Overview
- 1.1 The problem-solving cycle
- 1.2 Formulating a problem and decomposing it
- 1.3 Assumptions and what they cost you
- 1.4 Translating a situation into mathematics
- 1.5 Computing and interpreting a solution
- 1.6 Evaluating, refining and reporting
- 1.7 How to write a modelling report
- Review
Chapter 2: Networks and graphs
- Overview
- 2.1 Networks and their terminology
- 2.2 Connected, disconnected, directed and undirected graphs
- 2.3 Representing a network: adjacency matrices
- 2.4 Matrix algebra and the product of adjacency matrices
- 2.5 Trees and spanning trees
- 2.6 Minimum spanning trees: Kruskal's algorithm
- 2.7 Minimum spanning trees: Prim's algorithm
- 2.8 Shortest paths and Dijkstra's algorithm
- 2.9 Multi-stage problems and dynamic programming
- 2.10 Bellman's principle of optimality
- 2.11 Comparing the algorithms and choosing one
- Review
Chapter 3: Describing motion
- Overview
- 3.1 Displacement, velocity and acceleration
- 3.2 Motion with constant acceleration
- 3.3 Motion graphs
- 3.4 Differentiation and the chain rule
- 3.5 The fundamental theorem of calculus
- 3.6 Deriving the equations of motion by calculus
- 3.7 Projectile motion
- 3.8 Time of flight and range
- 3.9 Projectiles launched at an angle
- Review
Chapter 4: Forces and Newton's laws
- Overview
- 4.1 Forces and Newton's laws
- 4.2 Resultant force and equilibrium
- 4.3 Motion on a smooth horizontal plane
- 4.4 Motion on an inclined plane
- 4.5 Tension and connected particles
- 4.6 Pulleys
- 4.7 Newton's laws in vector form
- Review
Chapter 5: Friction and resistance
- Overview
- 5.1 Friction and the normal reaction
- 5.2 The coefficient of friction
- 5.3 Motion on rough planes
- 5.4 Statics on a rough plane
- 5.5 Drag proportional to a power of speed
- 5.6 Terminal velocity
- Review
Chapter 6: Momentum, impulse and collisions
- Overview
- 6.1 Momentum and impulse
- 6.2 Conservation of momentum
- 6.3 Collisions in one dimension
- 6.4 Newton's experimental law of restitution
- 6.5 Elastic and inelastic collisions
- 6.6 Collisions in two dimensions
- 6.7 Successive collisions and rebounds
- Review
Chapter 7: Circular motion and oscillation
- Overview
- 7.1 Angular velocity and period
- 7.2 Centripetal acceleration and force
- 7.3 Circular motion in context
- 7.4 Conical pendulum and banked tracks
- 7.5 Hooke's law and elastic strings
- 7.6 Simple harmonic motion
- Review
Chapter 8: Modelling a changing world
- Overview
- 8.1 Rates of change and differential equations
- 8.2 Exponential growth and decay
- 8.3 Recurrence relations as discrete models
- 8.4 First-order difference equations
- 8.5 Feedback and limited growth
- 8.6 The logistic model
- 8.7 Solving separable differential equations
- 8.8 Comparing a discrete and a continuous model
- Review
Chapter 9: The modelling project
- Overview
- 9.1 Reading the brief and choosing a problem
- 9.2 Building the model and stating assumptions
- 9.3 Testing the model against data
- 9.4 Iterating and improving
- 9.5 Communicating the result
- 9.6 Critical evaluation of models
- Review
About this book

Chapter 6: Momentum, impulse and collisions

Momentum, impulse, conservation of momentum, and modelling collisions as elastic or inelastic.

The chapter problem

What the crash test measures. You will model a collision using conservation of momentum, work out the impulse, and compare what an elastic model predicts with what an inelastic one does. By the end of the chapter you can say which model fits a real crash better and why.

Where it is used: Crash testing, sport, ballistics, and any impact you want to survive.

Your turn to connect it: Where would knowing what happens in a collision change how you design or do something?

Sections

6.1 Momentum and impulse — tier O · refs S3
6.2 Conservation of momentum — tier O · refs S3
6.3 Collisions in one dimension — tier O · refs S3
6.4 Newton's experimental law of restitution — tier O · refs S3
6.5 Elastic and inelastic collisions — tier O · refs S3
6.6 Collisions in two dimensions — tier H · refs S3
6.7 Successive collisions and rebounds — tier H · refs S3