A.5 · IB Physics
Special relativity
Near the speed of light, moving clocks run slow and moving lengths contract. Change the speed and watch the Lorentz factor γ set the scale of time dilation, length contraction and momentum.
Predict first
Start at the default speed of 0.600 c. If you raise the Speed, v slider to 0.800 c, what length does the 1.00 m moving ruler have?
L = L₀/γ and γ = 1/√(1 − v²/c²). At 0.800 c, γ = 1/√(1 − 0.64) = 1.67, so L = 1.00 ÷ 1.67 = 0.600 m.
Check it in the model: Move the Speed, v slider from 0.600 c to 0.800 c and read the Contracted length L in the readout.
moving object stationary observer's frame current speed
Δt = γ Δt₀ ; L = L₀ ÷ γ
p = γ m₀ v ; Ek = (γ − 1) m₀c²
c = 3.00 × 10⁸ m s⁻¹, m₀ = 1.00 kg
Try this
- Set v = 0.600 c and read γ. Check that a 1.00 s interval stretches to 1.25 s and a 1.00 m ruler shrinks to 0.800 m.
- Run the speed from 0.900 c to 0.990 c to 0.999 c. Why does γ barely change at everyday speeds but explode close to c on the graph?
- Compare the kinetic energy readout at 0.100 c with the value you get from ½m₀v². At which speed does the classical formula fail badly?
Exam tip: γ = 1/√(1 − v²/c²) is the same number for time dilation (Δt = γΔt₀) and length contraction (L = L₀/γ), so decide first which observer measures the proper time or proper length, then multiply or divide by γ.