A.5 · IB Physics
Light clock and time dilation
Watch a light clock on a train from the train and from the platform, change the train's speed, and see how the Lorentz factor stretches each tick. Then work out how much younger a travelling twin is after a round trip.
γ = 1 / √(1 − v²/c²)
Δt₀ = 2L / c (proper time)
(cΔt/2)² = L² + (vΔt/2)²
⇒ Δt = γ Δt₀
Δt₀ = 2L / c (proper time)
(cΔt/2)² = L² + (vΔt/2)²
⇒ Δt = γ Δt₀
Twin round trip
Try this
- Set v = 0.600c and count ticks: the train-frame clock completes 5 ticks while the platform view shows only 4 (γ = 1.25).
- Find the speed at which γ = 2.00 and check that the platform path per tick is twice 2L.
- For a 10 ly star, find the speed at which the travelling twin ages only 10 years on the round trip.
Common error: the proper time Δt₀ belongs to the observer for whom both events happen at the same place (the train clock), so it is always the shorter time – multiply it by γ, never divide.