Physic Labs

Theory of relativity

Relativistic momentum and energy, E = mc²

Watch how γ, energy E=γmc2E = \gamma mc^2, and momentum p=γmvp = \gamma m v vary with speed β = v/c. Verify the relation E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2 and the photon line E=pcE = pc on the plot.

Undergraduate

Equipment

  • «Bốn-động lượng» canvas: a rotatable 3D spacetime model (drag or arrow keys)
  • «Quan hệ định lượng» canvas: plot of relativistic quantities vs β, with a colored marker at the current value
  • «Tốc độ β = v/c» sliders (0–0.95), synchronized across both sections
  • «Tham số mô hình» (model parameter) slider; «Đặt lại góc nhìn» (reset view) and «Tạm dừng» (pause) buttons
  • Readouts «β, γ, x′ = γ(x − vt), t′ = γ(t − vx/c²)» and «E/(mc²) = γ; p/(mc) = γβ»

Procedure

  1. Measure γ vs speed

    Slide «Tốc độ β = v/c» through 0, 0.3, 0.6, 0.9, 0.95 and record «β = … γ = …» from the readout. For each β compute γ=1/1−β2\gamma = 1/\sqrt{1-\beta^2} and compare: β = 0.6 must give γ = 1.25, β = 0.95 gives γ ≈ 3.20. Note γ diverges as β → 1.

  2. Compare E/(mc²) and p/(mc)

    In the plot readout, record «E/(mc²) = γ» and «p/(mc) = γβ» for each β. Verify algebraically: (E/mc2)2−(p/mc)2=γ2(1−β2)=1\left(E/mc^2\right)^2 - \left(p/mc\right)^2 = \gamma^2(1-\beta^2) = 1 — the invariant rest mass. Example β = 0.8: γ = 5/3 ≈ 1.667, γβ = 4/3 ≈ 1.333, and 1.667² − 1.333² = 1.

  3. Read the photon line and the Lorentz transform

    On the «Bốn-động lượng» canvas read «Ánh sáng: s² = 0 | Vật có khối lượng: |v| < c» and the transform «x′ = γ(x − vt), t′ = γ(t − vx/c²)». Drag to rotate the model and inspect the timelike/lightlike worldlines; in section 2's plot, the m = 0 point sits exactly on E=pcE = pc — a photon carries energy and momentum but no rest mass.

  4. Predict the v → c limit

    Set β = 0.95 and re-read Δt/Δτ and L/L₀ in the readouts. Predict the trend as β → 1: γ → ∞ so it would take infinite energy to bring a massive body to c — which is why the slider stops at 0.95. Press «Đặt lại góc nhìn» to restore the standard view and «Tạm dừng» to inspect the plot point closely.

Simulation

Experiment history

In his 1905 paper «On the electrodynamics of moving bodies», Einstein derived the relativistic momentum p = γmv and kinetic energy (γ−1)mc², replacing the Newtonian expressions. That same year, in a short note to Annalen der Physik, he argued that a body emitting radiation loses mass by E/c² — the origin of the celebrated relation E = mc². In 1907–1908 Hermann Minkowski merged energy and momentum into a single four-vector: (E/c, p) is the four-momentum whose invariant norm is mc. This yields E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2, which automatically gives E = pc for m = 0 (the photon) and explains the displayed quantities γ and p/(mc) = γβ.

Related physicists

Albert Einstein

Related library topics

Relativistic momentum and energy, E = mc²