SME
University Physics III: Waves, Optics & Modern
Special relativity
Use this for “A probe passes Earth at 0.8c. Its own clock counts 12 μs between two flashes, and it is 40 m long at rest. What time and length do we measure?”
Diagram
Lorentz
factor:
γ
=
1/
√
1 −
β
2
=
1/
√
1 −
0.6
2
=
1.25
Moving clock: Δt = γΔt₀ = 1.25 × 2.2 μs = 2.75 μs
Moving length: L = L₀/γ = 100 m/1.25 = 80 m
Light moves at c for every observer, so the longer slanted path takes longer: a moving clock ticks slowly.
This is an example. Type your own values to start a new problem.
Δt (
)
=
γ
×
Δt₀ (
)
L (
)
=
L₀ (
)
γ
β
Speed as a fraction of c
example
Formulas
γ
=
1
÷
√
1 −
β
2
1.25
=
1
÷
√
1 −
0.6
2
Δt
=
γ
×
Δt₀
2.75 = 1.25 × 2.2
L
=
L₀
γ
80
=
100
1.25
Assumptions
•
Light moves at c in every inertial frame. With β = v ÷ c, the Lorentz factor γ = 1 ÷ √(1 − β²) is 1 at rest and grows without limit near c.
•
Δt₀ is the proper time, read on a clock that moves with the object; L₀ is the proper length, measured in the object’s own frame.
•
Moving past us, the clock runs slow, Δt = γΔt₀, and the object is shorter, L = L₀ ÷ γ; only lengths along the motion shrink.
Step-by-step
WE KNOW
β = 0.6, Δt₀ = 2.2 μs, L₀ = 100 m
FIND
Lorentz factor (γ), time we measure (Δt), length we measure (L)
Step 1 · Find Lorentz factor (γ)
Use γ = 1 ÷ √(1 − β²)
The
Lorentz
factor
for
this
speed:
1
at
rest,
growing
without
limit
as
β
nears
1.
γ
=
1
÷
√
1 −
0.6
2
γ
=
1
÷
√
1 − 0.36
γ
=
1
÷
√
0.64
γ
=
1
0.8
γ
=
1.25
Step 2 · Find time we measure (Δt)
Use Δt = γ × Δt₀
The
moving
clock’s
light
zigzags
a
longer
path
at
c,
so
each
tick
lasts
γ
times
longer
as
we
measure
it.
Δt = 1.25 × 2.2
Δt = 2.75 μs
Step 3 · Find length we measure (L)
Use L = L₀ ÷ γ
Along
the
motion
the
object
is
shorter
by
the
same
factor
γ
.
L
=
100
1.25
L
=
80
m
Check
Put the numbers back in. Both sides should match.
1.25
=
1
÷
√
1 −
0.6
2
✓
2.75 = 1.25 × 2.2 ✓
80
=
100
1.25
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
Relativistic velocity addition
Use this for “A ship moving at 0.7c fires a probe forward at 0.5c relative to the ship. How fast does the probe move as seen from the planet?”
Relativistic energy and momentum
Use this for “A proton (rest energy 938.3 MeV) moves at 0.8c. Find its total energy, its kinetic energy and its momentum in MeV/c.”
The Lorentz transformation
Use this for “A rocket passes Earth at 0.8c. An event happens 1,200 m away along its path, 2,000 m of ct after they meet. Where and when is it in the rocket’s frame?”