Every observer measures the same speed of light in a vacuum, c = 3.00 × 108 m s−1, however fast they are moving. Two results follow: a moving clock runs slow, and a moving object is shorter. Set a speed below and watch both happen.
Concept 1
Time dilation
Time dilation
t′ = t√1 − (vc)2
as on the relationships sheet
t′
time measured by an observer the clock moves past
s
t
time measured beside the clock, moving with it
s
v
speed of the clock
m s−1
c
speed of light, 3.00 × 108
m s−1
Light clocks
Set a speed, then compare the two clocks. 0.995c is about the speed of the muons in the worked example below.
Predict how the ticks of the moving clock compare with the clock at rest beside you.
A light clock is a pulse of light bouncing between two mirrors. One tick is one trip up and back.
To you, the moving clock's light follows a longer, slanted light path, still at c, so each tick takes longer than a tick of the clock at rest.
You measure a moving clock running slow. This is time dilation.
Time on your clock, t′
Time on the moving clock, t
Ticks: yours0
Ticks: moving clock0
t′ is always bigger than t. If your answer is smaller, you multiplied when you should have divided.
Presets
Clocks
Worked examples
Substitute & solve
A muon lives for 2.2 μs, measured in its own frame. It travels towards Earth at 0.995c. Calculate its lifetime measured by an observer on Earth.
Convert first
t = 2.2 × 10−6 s μs → s
v/c = 0.995 read off 0.995c
t′=t√1 − (vc)2
t′=2.2 × 10−6√1 − 0.9952numbers in
t′=2.2 × 10−6 ÷ 0.0999square, take from 1, root
t′=2.2 × 10−5 sten times longer
Rearrange — numbers in first
A clock on a spacecraft records a journey time of 30 minutes. Observers on Earth measure the same journey as 50 minutes. Calculate the speed of the spacecraft.
t′=t√1 − (vc)2
50=30√1 − (vc)2numbers in first; minutes cancel
√1 − (vc)2=30 ÷ 50 = 0.60now rearrange
1 − (vc)2=0.36square both sides
(vc)2=0.64
v=0.80 × 3.00 × 108root, then × c
v=2.4 × 108 m s−1
Practice 1
Concept 2
Length contraction
Length contraction
l′ = l√1 − (vc)2
as on the relationships sheet
l′
length measured by an observer the object moves past
m
l
length measured at rest beside the object
m
v
speed of the object
m s−1
c
speed of light, 3.00 × 108
m s−1
Rocket and ruler
The same speed as the light clocks: change it in either place.
At 0.60c a rocket is measured 0.80 times as long as at rest. Predict what happens to its height.
The rocket at rest has length l, measured by someone beside it.
To an observer it is moving past, it is shorter: length l′. This is length contraction. Its height does not change.
The same factor as time dilation, √1 − (v/c)2: multiply for lengths, divide for times.
Factor √1 − (v/c)2
Moving length
l′ is always smaller than l.
Presets
Worked examples
Substitute & solve
A spacecraft is 120 m long when at rest. It passes Earth at 1.8 × 108 m s−1. Calculate its length measured by an observer on Earth.
Convert first
v/c = 1.8 × 108 ÷ 3.00 × 108speed as a fraction of c
v/c = 0.60
l′=l√1 − (vc)2
l′=120 × √1 − 0.602numbers in
l′=120 × 0.80
l′=96 mshorter, as it should be
Rearrange — numbers in first
A probe is 50 m long at rest. An observer measures its length as 40 m as it flies past. Calculate the speed of the probe.
l′=l√1 − (vc)2
40=50 × √1 − (vc)2numbers in first
√1 − (vc)2=40 ÷ 50 = 0.80metres cancel
1 − (vc)2=0.64square both sides
(vc)2=0.36
v=0.60 × 3.00 × 108root, then × c
v=1.8 × 108 m s−1
Practice 2
Go further
A journey to a star
The traveller and Earth disagree about the time and the distance, but they agree on what happens. Earth sees a slow clock. The traveller sees a shorter distance. Both give the same reading on the traveller's clock.
The journey
Show both points of view
Check yourself
Multiple choice
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Words to know
Light clock
Light bouncing between two mirrors. One trip up and back is one tick.
Time dilation
A moving clock is measured to run slow: t′ is longer than t.
Length contraction
A moving object is measured to be shorter along its direction of motion: l′ is shorter than l.
Frame of reference
The point of view an observer measures from, and moves with.
