Interactive module First-year engineering
Curvilinear Motion
A thrown ball, a car on a winding road and a robot gripper all move along curved paths. This module teaches you to describe that motion two ways: with fixed rectangular components \((x, y)\) and with path components \((n, t)\) that ride with the particle. Nine short lessons, with animated figures and questions that check your work, take you from position vectors to highway design.
- 9 lessons
- about 5 h of lessons
- Works offline
Learning outcomes
By the end of the module you can:
- Explain what changes when a path curves, and why a particle at constant speed on a curve is accelerating. (Lesson 1)
- Define position, displacement, velocity and acceleration as vectors, and explain why \(\vvec\) is tangent to the path. (Lesson 2)
- Compute velocity and acceleration in rectangular components from \(x(t)\), \(y(t)\), from a path \(y = f(x)\), or by integrating \(\avec(t)\). (Lesson 3)
- Solve projectile problems: time of flight, maximum height, range, launches from a height and aiming at a target. (Lesson 4)
- Describe motion in path coordinates with \(\vvec = v\,\et\) and \(\avec = \dot v\,\et + (v^2/\rho)\,\en\), including circular motion. (Lessons 5–6)
- Find the radius of curvature of a path \(y = f(x)\) or of a motion \(x(t), y(t)\). (Lesson 7)
- Convert between rectangular and path components, choose the better system for a problem, and apply \(a_n = v^2/\rho\) to design limits. (Lessons 8–9)
How to use this module
Before you start: you should be comfortable with 2D vectors (components, magnitude, the dot product), sine and cosine of angles in degrees and radians, derivatives including the product and chain rules, and simple integrals. Rectilinear (straight-line) kinematics is reviewed in Lesson 1.
Work through the lessons in order. The times allow for working the examples on paper and doing the checks. A suggested plan in four sittings:
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Lessons 1–385 min
From straight lines to curves, the motion vectors, and rectangular components.
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Lessons 4–570 min
Projectile motion, then path coordinates: \(\et\), \(\en\) and the osculating circle.
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Lessons 6–765 min
Tangential and normal acceleration, and the radius of curvature.
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Lessons 8–9, Practice Lab and Self-Check Quiz100 min
Connecting the two systems and engineering applications, about 30 min of practice, then the 35-minute quiz.
About 5½ h in total:290 min of lessons, 30 min of practice, 35 min for the quiz
Get the most out of it
- Keep pencil and paper handy
Try each worked example yourself before you open its solution, and sketch the figures.
- Do the checks
Every lesson has “Check your understanding” questions with hints and full solutions. You can type answers like
sqrt(61)or25^2/150. - Practice until it is routine
The Practice Lab generates unlimited problems, each with a worked solution.
- Test yourself at the end
The Self-Check Quiz gives a score, solutions and a results page you can print or save as a PDF.
- Track your progress
Tick “Mark this lesson complete” at the end of each lesson. A ✓ appears here and in Contents; it is saved in this browser only.
- Keep the formulas close
Print the Formula Sheet, and look up terms in the Glossary.
Lessons
Each lesson has interactive figures, worked examples and questions with instant feedback.
Practice, tools and reference
Use these alongside the lessons, or on their own when you revise.
Requirements
Any modern browser. Works offline.
- A current version of Chrome, Edge, Firefox or Safari, with JavaScript on. A laptop or desktop screen works best; the pages also fit tablets and phones.
- No internet connection, installation or account. Everything is inside this folder.
- Your progress is saved in this browser on this device. Private or incognito windows do not keep it.
Something not working? Read README.txt in the module folder for how to open the module and fix common problems.