← Mechanics

Reasoning about a change without calculating

An Excellence-level question type that appears without fail: "without calculating, compare/predict/justify what happens if...". These are graded on the REASONING, not a number.

The method for a "without calculating" question

  • Identify which formula governs the quantity being asked about (e.g. E_k = ½mv² for compression distance, v = 2πr/T for period).
  • Identify exactly which variable in that formula the question changes (e.g. "twice the speed" doubles v).
  • State the PROPORTIONAL effect that change has on the answer, using the formula's structure — a SQUARED term means the effect is squared too, not doubled.
  • Conclude with the actual comparison the question asked for (bigger/smaller, by what factor), not just a repeat of the formula.
Worked example Doubling speed and its effect on stopping distance A player's kinetic energy transfers entirely into compressing a tackle bag (elastic PE, E_e = ½kx²). Without calculating, state and justify what happens to the compression distance if the player moves at TWICE the speed. The governing relationship: all of the kinetic energy ½mv² becomes elastic PE ½kx², so ½mv² = ½kx², meaning x² ∝ v². Since x² is proportional to v², x itself is proportional to v (taking the square root of both sides). Doubling v therefore doubles x directly, since x ∝ v — NOT a factor of four. ✓ The compression distance doubles. Note this is a genuinely different result from a question about ENERGY itself, where doubling v would quadruple the energy (since E_k ∝ v²) — the "without calculating" step is entirely about correctly tracking which power of the changed variable survives into the quantity actually being asked about. ✕

Squaring the wrong quantity is the single most common error here

When a formula contains a squared term (E_k = ½mv², E_e = ½kx²), doubling the variable that is INSIDE the square (v or x) affects the ENERGY by a factor of four. But if the QUESTION asks about the variable inside the square itself (e.g. "what happens to x"), and x is related to v by a square-then-root chain, the relationship between x and v may end up being direct, not squared. Always write out exactly which quantity the question asks about and trace the actual proportionality chain — do not assume every doubling produces a squared effect.

Practise just this