The γ(β) Curve Explorer

Source

Collection: Wiki SR simulations (v1) — SuperSTEM special-relativity engine build.
Concept: The Lorentz factor γ = 1/√(1−β²) stays near 1 for ordinary speeds and diverges to ∞ only as β → 1.
Serves: Intuition for why relativistic effects are invisible in daily life yet become enormous near light speed: γ − 1 grows only as ½β² for small β, then blows up near β = 1.
Inquiry: At what fraction of light speed does a moving clock run at half rate (γ = 2)? Why does the curve look flat for so long before it explodes?
Misconception: That γ climbs steadily / linearly with β, so half of light speed already gives a large γ. In fact γ = 2 needs β ≈ 0.866 (87% of c).

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Misconception is structural, not a text card	The "half of c gives γ = 2" error is a wrong prediction choice; its rebuttal (a directional hint) appears only if the student picks it.
D6/D7	Guided inquiry reveals in stages	Sidebar step-cards run predict → explore → verify → push toward β → 1 → resolve, letting the student drag the marker to find γ = 2 for themselves.
A2	Fewest objects	Opens with one curve and one draggable marker; advanced numbers (v, γ − 1) hide behind a toggle, reference speeds behind another.
C	Every number earns its place	β, γ and the local-slope verdict are each read or predicted by a step; discrete γ-max scales {5, 50, 7000} let divergence be seen without auto-rescale hiding it.
Phys	Physics honesty — a function plot is not a particle	β is a static value the student sets by dragging the marker; nothing auto-sweeps β over time (which would imply an accelerating frame).

Speed

γ vs βdrag the marker along the curve

jet GPS ISS muon LHC

Magnifying insetneighbourhood of the marker

Guided Inquiry

1 · Predict where γ = 2

A moving clock runs at half rate when γ = 2. At roughly what fraction of light speed β does that happen?

2 · Explore the curve

Drag the amber marker along the curve (or type a β in the sidebar). Watch how flat γ stays for small β, then how sharply it climbs. Find the β where the γ readout reads 2.000.

3 · Why does it look flat so long?

Across the whole first half of the axis (β up to ≈ 0.5), γ barely leaves 1. Why?

4 · Push β toward 1

Drag the marker close to the right edge (β → 1) and switch the γ-max scale up to 50, then 7 000. As β approaches 1, γ …

5 · The whole story

γ hugs 1 for ordinary speeds (γ − 1 ≈ ½β²), reaches 2 only at β ≈ 0.866, and diverges to ∞ as β → 1. That is why relativity hides at everyday speeds yet dominates near light speed. Turn on ∑ Formal (top bar) for the algebra, or keep dragging the marker and snapping to real objects. Hit Finish to tuck this away and explore freely.

Controls

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γ-axis

γ-max

β = v / c

Snap to a real object

Readouts (lab / Earth frame)

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β = v/c—

γ—

dγ/dβ—

Reference speeds

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The Lorentz factor

It equals 1 at β = 0 and → ∞ as β → 1.

Small-β expansion

For everyday speeds (β ≪ 1) the departure from 1 grows only as β², which is why the curve looks perfectly flat there.

Where γ = 2

γ = 2 (clocks at half rate) requires β = √3⁄2 ≈ 0.8660 — already 87% of light speed.