SME
Grade 11 Science · Problem type
Moving clocks run slow
Use this for “A spaceship passes at 0.6c. A clock on board ticks 10 s. How long does that take as we see it? How long is the 100 m ship to us?”
Diagram
Lorentz
factor:
γ
=
1/
√
1 −
β
2
=
1/
√
1 −
0.6
2
=
1.25
Moving clock: Δt = γΔt₀ = 1.25 × 10 s = 12.5 s
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.
γ
=
1
1
−
β
2
Δt₀
Time on the moving clock
example
Δt
Time as we measure it
calculated
L₀
Length at rest
example
L
Length as we measure it moving
calculated
Formulas
γ
=
1/
√
1 −
β
2
1.25
=
1/
√
1 −
0.6
2
Δt
=
γ
×
Δt₀
12.5 = 1.25 × 10
L
=
L₀
γ
80 = 100/1.25
Assumptions
•
Light moves at c for every observer, however they move.
•
γ = 1/√(1 − β²), with β = v/c. It is 1 at rest and grows without limit near c.
•
Δt₀ and L₀ are measured beside the clock or rod; moving past us, Δt = γΔt₀ and L = L₀/γ.
Step-by-step
WE KNOW
β = 0.6, Δt₀ = 10 s, L₀ = 100 m
FIND
Lorentz factor (γ), time as we measure it (Δt), length as we measure it moving (L)
Step 1 · Find Lorentz factor (γ)
Use γ = 1/√(1 − β²)
The Lorentz factor for this speed.
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 as we measure it (Δt)
Use Δt = γ × Δt₀
The
moving
clock’s
tick,
stretched
by
γ
.
Δt = 1.25 × 10
Δt = 12.5 s
Step 3 · Find length as we measure it moving (L)
Use L = L₀/γ
Shorter
along
the
motion
by
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
✓
12.5 = 1.25 × 10 ✓
80 = 100/1.25 ✓
Topic check
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