Bertozzi's Ultimate Speed

Plate 30 speaks for itself: “Push harder; it barely goes faster.”

The Physics

Bertozzi timed electrons over an 8.4 m flight path at kinetic energies from 0.5 to 15 MeV. Thirty times more energy barely changed the flight time: kinetic energy and momentum grow without limit, but the speed only creeps toward c.

v = c √(1 − 1/γ²), γ = 1 + K/(m_e c²)

• v: electron speed, m s⁻¹
• c: speed of light, m s⁻¹
• γ: Lorentz factor, dimensionless
• K: kinetic energy, J (quoted in MeV)
• m_e c²: electron rest energy, 0.51099895 MeV

MIT, 1964: electrons, 0.5 to 15 MeV. 8.4 m in 32.3 ns, then 28.0 ns.

Honest caveat: The extra push goes into energy and momentum (p = γ m v), not into a growing mass; the old idea of 'relativistic mass' is avoided here. The 4.5 and 15 MeV points involve acceleration along the path.

Sources

• W. Bertozzi, Am. J. Phys. 32, 551 (1964).

On the poster

Plate 030, “Bertozzi's Ultimate Speed”: the pixel-art plate with its slogan “Push harder; it barely goes faster.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.