---
title: "Time Dilation and Length Contraction"
description: "For constant relative velocity, a moving clock records less elapsed time than the measuring frame's coordinate clocks. Moving lengths measured simultaneously are shorter along the motion. Atmospheric "
canonical: https://lightmysky.com/learn/science/time-dilation-and-length-contraction-mt_7tpKiFWeG0
source: https://lightmysky.com/learn/science/time-dilation-and-length-contraction-mt_7tpKiFWeG0.md
retrieved: 2026-11-12
---

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# Time Dilation and Length Contraction

For constant relative velocity, a moving clock records less elapsed time than the measuring frame's coordinate clocks. Moving lengths measured simultaneously are shorter along the motion. Atmospheric muon survival supports these predictions.

Subject: Science · Area: Quantum & Modern Physics · Ages 18 to 19
Page: https://lightmysky.com/learn/science/time-dilation-and-length-contraction-mt_7tpKiFWeG0

## Ready when they can

- Derives the time dilation factor from a light clock
- Applies dilation and contraction consistently to the same situation from both frames
- Explains the survival of atmospheric muons from either frame and gets the same answer

## Lesson: Measuring moving clocks and lengths

A ship moves at constant speed v past the ground. Its light-clock mirrors are separated perpendicular to v. One tick is emission, reflection and return to the starting mirror. The ship records proper time Δτ between emission and return at one place. Ground time Δt refers to the same events after correcting signal delays. Light in a vacuum always travels at the same speed c in every inertial frame, whatever the motion of its source or the observer. At that speed c, light takes longer to travel the longer diagonal ground path.

Let D be the mirror gap, which is transverse and unchanged between these frames. In a ground half-tick, light travels cΔt/2 while the ship moves vΔt/2. These form a right triangle with legs D and vΔt/2. Pythagoras gives (cΔt/2)^2=D^2+(vΔt/2)^2. The ship has 2D=cΔτ, so D=cΔτ/2. Substitute and collect terms: (1−v²/c²)Δt²=Δτ². Take the positive square root to get Δt=Δτ/√(1−v²/c²).

*(drawing: These cells show algebra, not the moving-mirror triangle. Δτ is ship tick time; Δt is ground tick time.)*

The Lorentz factor is γ=1/√(1−v²/c²), for 0≤v<c. At v=0, γ=1; it grows with speed. For one clock's events, Δt=γΔτ. With γ=3, a 4 s clock interval gives 12 s in the other frame. Each frame measures the other's clocks as slow after correcting signal delays. The reciprocal comparisons use different simultaneous-event choices.

Proper length L0 is measured with the object at rest. For a moving rod, locate both endpoints at the same time in the measuring frame using its synchronized clocks. Along relative motion, L=L0/γ. With γ=3, a 9 m rest length is measured as 3 m. Perpendicular lengths are unchanged. This is a comparison between frames, not physical crushing.

Atmospheric muons form in cosmic-ray interactions and decay unpredictably. Survival describes a fraction of a group. The ground measures their moving decay clock's half-life as γ times its rest half-life. Fewer half-lives pass during the same ground travel time, so more survive than using the rest half-life as ground time would predict. No muon needs to exceed c; not every muon survives.

Let H be the vertical gap between two levels fixed in the ground frame, so H is their proper separation. Muons travel straight down along H at constant speed v. Ground flight time is H/v; the muon frame has gap H/γ and flight time H/(γv). For rest half-life h, ground half-life is γh; muon half-life is h. Divide each frame's flight time by its half-life. Equal counts give equal survival fractions: one half-life leaves 1/2 on average, two leave 1/4.

**Example.** A half-life is the time for half a large group to decay on average. Two half-lives leave one quarter. Use a rounded muon rest half-life of 1.5 μs, γ=3 and ground flight time 4.5 μs. The ground half-life is 3×1.5=4.5 μs. The muon-frame flight time is 4.5÷3=1.5 μs. Each frame counts one half-life and predicts one half survives. This model ignores energy loss and speed changes.

**Recap.** For one clock's events, Δt=γΔτ. A moving length measured simultaneously is L0/γ along motion. Both frames predict the same muon survival fraction.

*Lesson, questions and study cards written for this learning dataset. OpenStax relativity chapters were consulted for facts.*

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [The Postulates of Special Relativity and the Loss of Simultaneity](https://lightmysky.com/learn/science/the-postulates-of-special-relativity-and-the-loss-of-simultaneity-mt_kZqVQ18bbc)