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.
Length along the motion
Explore the relationship
Both ends must be measured simultaneously in the observer’s frame. Transverse lengths are unchanged.
Tap or focus a plotted point to read its values.
View plotted data as a table
| Series | Speed / c | Measured parallel length / m |
|---|---|---|
| Measured parallel length / m | 0 | 5 |
| Measured parallel length / m | 0.0158333 | 4.99937 |
| Measured parallel length / m | 0.0316667 | 4.99749 |
| Measured parallel length / m | 0.0475 | 4.99436 |
| Measured parallel length / m | 0.0633333 | 4.98996 |
| Measured parallel length / m | 0.0791667 | 4.98431 |
| Measured parallel length / m | 0.095 | 4.97739 |
| Measured parallel length / m | 0.110833 | 4.9692 |
| Measured parallel length / m | 0.126667 | 4.95973 |
| Measured parallel length / m | 0.1425 | 4.94897 |
| Measured parallel length / m | 0.158333 | 4.93693 |
| Measured parallel length / m | 0.174167 | 4.92358 |
| Measured parallel length / m | 0.19 | 4.90892 |
| Measured parallel length / m | 0.205833 | 4.89294 |
| Measured parallel length / m | 0.221667 | 4.87561 |
| Measured parallel length / m | 0.2375 | 4.85694 |
| Measured parallel length / m | 0.253333 | 4.8369 |
| Measured parallel length / m | 0.269167 | 4.81547 |
| Measured parallel length / m | 0.285 | 4.79264 |
| Measured parallel length / m | 0.300833 | 4.76838 |
| Measured parallel length / m | 0.316667 | 4.74268 |
| Measured parallel length / m | 0.3325 | 4.71552 |
| Measured parallel length / m | 0.348333 | 4.68685 |
| Measured parallel length / m | 0.364167 | 4.65667 |
| Measured parallel length / m | 0.38 | 4.62493 |
| Measured parallel length / m | 0.395833 | 4.59161 |
| Measured parallel length / m | 0.411667 | 4.55667 |
| Measured parallel length / m | 0.4275 | 4.52008 |
| Measured parallel length / m | 0.443333 | 4.48178 |
| Measured parallel length / m | 0.459167 | 4.44175 |
| Measured parallel length / m | 0.475 | 4.39993 |
| Measured parallel length / m | 0.490833 | 4.35627 |
| Measured parallel length / m | 0.506667 | 4.31071 |
| Measured parallel length / m | 0.5225 | 4.2632 |
| Measured parallel length / m | 0.538333 | 4.21366 |
| Measured parallel length / m | 0.554167 | 4.16203 |
| Measured parallel length / m | 0.57 | 4.10822 |
| Measured parallel length / m | 0.585833 | 4.05216 |
| Measured parallel length / m | 0.601667 | 3.99374 |
| Measured parallel length / m | 0.6175 | 3.93285 |
| Measured parallel length / m | 0.633333 | 3.8694 |
| Measured parallel length / m | 0.649167 | 3.80323 |
| Measured parallel length / m | 0.665 | 3.73422 |
| Measured parallel length / m | 0.680833 | 3.66219 |
| Measured parallel length / m | 0.696667 | 3.58697 |
| Measured parallel length / m | 0.7125 | 3.50836 |
| Measured parallel length / m | 0.728333 | 3.42611 |
| Measured parallel length / m | 0.744167 | 3.33997 |
| Measured parallel length / m | 0.76 | 3.24962 |
| Measured parallel length / m | 0.775833 | 3.15469 |
| Measured parallel length / m | 0.791667 | 3.05477 |
| Measured parallel length / m | 0.8075 | 2.94934 |
| Measured parallel length / m | 0.823333 | 2.83779 |
| Measured parallel length / m | 0.839167 | 2.71937 |
| Measured parallel length / m | 0.855 | 2.59314 |
| Measured parallel length / m | 0.870833 | 2.45789 |
| Measured parallel length / m | 0.886667 | 2.31205 |
| Measured parallel length / m | 0.9025 | 2.15345 |
| Measured parallel length / m | 0.918333 | 1.97904 |
| Measured parallel length / m | 0.934167 | 1.78418 |
| Measured parallel length / m | 0.95 | 1.56125 |
Explore: Increase speed while keeping proper length fixed. Compare parallel length with the unchanged transverse dimension.
A rod of proper length 10 m moves lengthwise at 0.60c.
- √(1 − ) = 0.80.
- The measured length is 8.0 m.
Assumed knowledge
Time dilation and inertial frames.
Learning checkpoints & source
YOUR LEARNING CHECKPOINT- Identify proper length and length contraction.
- Explain frame-dependent simultaneity.