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Worked Examples

Mechanics is a subject where the hard part is almost never the algebra — it is deciding which conservation law applies to which stage of a problem. Each example below is worked end to end with that decision made explicit, and each one contains at least one place where the obvious move is wrong.

  • The ballistic pendulum — two stages, two different conservation laws, and the classic mistake of using the wrong one first
  • The rolling race — why a sphere beats a disc beats a hoop, why mass and radius do not matter at all, and why a frictionless block beats them all
  • Hohmann transfer — getting from low orbit to geostationary, worked with vis-viva
  • Beats and tuning a guitar — why musicians listen for the throb rather than the pitch

The collision lab reproduces the first example's momentum and energy bars; the orbit sandbox runs the third with a real integrator; the wave studio makes the fourth audible on screen.

How to use these

Read the problem statement, close the page, and try it. The value is in noticing which step you would have got wrong — every example flags one, and they are the same steps that cost marks in exams.