weirdfacts
Year 12 / Special relativity

Measure both ends at once.

Proper length is measured in the object’s rest frame. An observer seeing the object move measures a shorter length parallel to the motion, provided both endpoints are measured simultaneously in that observer’s frame. Perpendicular dimensions are unchanged.

Events simultaneous in one inertial frame need not be simultaneous in another. This resolves apparent contradictions such as a moving ladder fitting into a shorter barn: the timing of the two door events changes between frames. The travelling twin’s turnaround changes inertial frames, so the twins’ histories are not symmetric.

INTERACTIVE MODEL

Length along the motion

READ THE GRAPH

Explore the relationship

Both ends must be measured simultaneously in the observer’s frame. Transverse lengths are unchanged.

Measured parallel length / m01.3132.6253.9385.2500.23750.4750.71250.95Speed / c
Measured parallel length / m

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Explore the relationship
SeriesSpeed / cMeasured parallel length / m
Measured parallel length / m05
Measured parallel length / m0.01583334.99937
Measured parallel length / m0.03166674.99749
Measured parallel length / m0.04754.99436
Measured parallel length / m0.06333334.98996
Measured parallel length / m0.07916674.98431
Measured parallel length / m0.0954.97739
Measured parallel length / m0.1108334.9692
Measured parallel length / m0.1266674.95973
Measured parallel length / m0.14254.94897
Measured parallel length / m0.1583334.93693
Measured parallel length / m0.1741674.92358
Measured parallel length / m0.194.90892
Measured parallel length / m0.2058334.89294
Measured parallel length / m0.2216674.87561
Measured parallel length / m0.23754.85694
Measured parallel length / m0.2533334.8369
Measured parallel length / m0.2691674.81547
Measured parallel length / m0.2854.79264
Measured parallel length / m0.3008334.76838
Measured parallel length / m0.3166674.74268
Measured parallel length / m0.33254.71552
Measured parallel length / m0.3483334.68685
Measured parallel length / m0.3641674.65667
Measured parallel length / m0.384.62493
Measured parallel length / m0.3958334.59161
Measured parallel length / m0.4116674.55667
Measured parallel length / m0.42754.52008
Measured parallel length / m0.4433334.48178
Measured parallel length / m0.4591674.44175
Measured parallel length / m0.4754.39993
Measured parallel length / m0.4908334.35627
Measured parallel length / m0.5066674.31071
Measured parallel length / m0.52254.2632
Measured parallel length / m0.5383334.21366
Measured parallel length / m0.5541674.16203
Measured parallel length / m0.574.10822
Measured parallel length / m0.5858334.05216
Measured parallel length / m0.6016673.99374
Measured parallel length / m0.61753.93285
Measured parallel length / m0.6333333.8694
Measured parallel length / m0.6491673.80323
Measured parallel length / m0.6653.73422
Measured parallel length / m0.6808333.66219
Measured parallel length / m0.6966673.58697
Measured parallel length / m0.71253.50836
Measured parallel length / m0.7283333.42611
Measured parallel length / m0.7441673.33997
Measured parallel length / m0.763.24962
Measured parallel length / m0.7758333.15469
Measured parallel length / m0.7916673.05477
Measured parallel length / m0.80752.94934
Measured parallel length / m0.8233332.83779
Measured parallel length / m0.8391672.71937
Measured parallel length / m0.8552.59314
Measured parallel length / m0.8708332.45789
Measured parallel length / m0.8866672.31205
Measured parallel length / m0.90252.15345
Measured parallel length / m0.9183331.97904
Measured parallel length / m0.9341671.78418
Measured parallel length / m0.951.56125
Measured parallel length / m: 5

Explore: Increase speed while keeping proper length fixed. Compare parallel length with the unchanged transverse dimension.

L=L01v2/c2=L0γL=L_0\sqrt{1-v^2/c^2}=\frac{L_0}{\gamma}
WORKED EXAMPLE

A rod of proper length 10 m moves lengthwise at 0.60c.

  1. √(1 − 0.6020.60^{2}) = 0.80.
  2. The measured length is 8.0 m.
Assumed knowledge

Time dilation and inertial frames.

Learning checkpoints & sourceYOUR LEARNING CHECKPOINT
  • Identify proper length and length contraction.
  • Explain frame-dependent simultaneity.
QCAA Physics · Unit 4 · Special relativity
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