Length Contraction vs. Speed

Source

Collection: Wiki SR simulations (v1) — SuperSTEM special-relativity engine build.
Concept: A rod of rest length L₀ moving at speed v is measured shorter along its direction of motion: L = L₀√(1−β²) = L₀/γ, with β = v/c.
Serves: The shape of the contraction law: L/L₀ is nearly flat at low speed and collapses only as β → 1 — the whole effect hides right up against the speed of light.
Inquiry: As v climbs from 0 toward c, does the measured length fall steadily, drop fast then level off, or stay near L₀ until the very end?
Misconception: That length trades off linearly with speed (a straight line down), or that contraction saturates quickly to a fixed shorter length.

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Misconception is structural, not a text card	The "straight-line" and "drops-then-levels-off" errors are wrong prediction choices; each wrong pick overlays its predicted shape as a dashed red curve so the student sees it diverge from the real √(1−β²) curve.
D6/D7	Guided inquiry reveals in stages	Sidebar step-cards run predict → observe the rod → read the curve → verify the shape → resolve; the theoretical curve is revealed as the student needs it.
J	Invariant panels are scoped to invariance sims	This is a contraction sim, not an invariance sim, so no interval/invariant readout is shown — only β, γ, L, and L/L₀.
A2	Fewest objects	Opens with one rest rod and one moving rod plus the single L/L₀-vs-β curve; γ, the classical overlay, and the theory curve are opt-in toggles.
C	Every number earns its place	β, γ, L, and L/L₀ are each read or compared by a step; the visited-points trail lets the student build the curve by exploring.
Phys	No false animation of β over time	Velocity is a static slider the student sets; the only motion is the rod translating past the observer at that constant speed (an inertial worldline), and the plot marker is tied to the slider, not a dot sweeping the curve.

Length Contraction

Speed

The rodrest length vs. measured (moving) length

moving rod (contracted) would-be classical length

L / L₀ vs. βthe contraction law √(1−β²)

L/L₀ = √(1−β²) points you've visited your prediction γ classical

Guided Inquiry

1 · Predict the shape

A rod of rest length L₀ flies past you. As v goes from 0 up toward c, how does the length L you measure behave?

2 · Watch the rod fly past

Drag the speed slider in the sidebar, then press ▶ in the top bar. The top bar is the rod's rest length L₀; the bottom bar is the length you measure as it flies past. The dashed gold outline marks the length it would have with no contraction.

3 · Read the curve

The right plot draws the theoretical L/L₀ = √(1−β²) in cyan. Sweep the slider slowly and leave a trail of green visited points. Notice how much you can raise β before L/L₀ drops noticeably — the curve barely leaves 1 until β is large.

4 · Where does the effect live?

At which speed does the measured length first fall below 90% of L₀?

5 · Resolve

Length contracts only along the direction of motion — the rod's height is identical in both frames, which is why the two bars always share the same height. The law L = L₀√(1−β²) = L₀/γ is flat at low speed and dives to zero as β → 1. Toggle γ and the classical overlay, open ∑ Formal for the algebra, then hit Finish to explore freely.

Controls

▾

β 0.000c

00.999c

Classical

γ curve

L/L₀ curve

Readouts

▾

β = v/c—

L₀ (rest)1.000 m

L (measured, ground frame)—

L / L₀—

Length contracts only along the direction of motion. The rod's vertical (transverse) extent is identical in both frames — that is why the two rods share the same height, always.

Speed parameter

Lorentz factor

Length contraction