Electron Speed Calculator
Calculate electron speed from kinetic energy — complete with the non-relativistik v = √(2E/m) and relativistik v = c·√(1 − 1/γ²), plus β (fraction of c), the Lorentz factor γ, and an honest marker of when the relativistic regime is required applies. It also works in reverse: energy from speed.
- v relativistik
- v klasik √(2E/m)
- β = v/c
- γ (Lorentz)
- Klasik melebih-lebihkan
- E relativistik (γ−1)mc²
- E klasik ½mv²
- β = v/c
- γ (Lorentz)
Reference table: energy → speed (relativistic)
| E kinetik | v relativistik (m/s) | β = v/c | γ | Klasik melebihi |
|---|---|---|---|---|
| 1 eV | 5,930961 × 10^5 | 0,0019784 | 1,000001957 | 0,0001% |
| 100 eV | 5,930099 × 10^6 | 0,0197807 | 1,000195695 | 0,0147% |
| 1 keV | 1,872790 × 10^7 | 0,0624695 | 1,001956951 | 0,1467% |
| 10 keV | 5,845521 × 10^7 | 0,1949856 | 1,019569512 | 1,4618% |
| 100 keV | 1,643525 × 10^8 | 0,5482209 | 1,195695118 | 14,1168% |
| 1 MeV | 2,821285 × 10^8 | 0,9410792 | 2,956951184 | 110,222% |
| 10 MeV | 2,994380 × 10^8 | 0,9988176 | 20,56951184 | 526,352% |
Figures verified against the Decimal-50 oracle (16 vectors, frozen). The 10 keV range is typical of X-ray tubes & SEM/TEM; above 255.5 keV (= mec²/2) the classical formula predicts v > c.
Step-by-step example: E = 100 eV
Metodologi & konstanta
- Electron mass me = 9,1093837015×10⁻³¹ kg (CODATA 2018).
- Elementary charge e = 1.602176634×10⁻¹⁹ C — exact (2019 SI definition); 1 eV = e × 1 V.
- Speed of light c = 299792458 m/s — exact.
- Electron rest energy mec² = 510998,95 eV = 510,999 keV.
- Regime: classical when E < ~10 keV (β ≤ 0.2); above that use relativistic.
- All figures on this page are verified against the Decimal-50 oracle (16 vectors, frozen).
- #1607 (this page) = electron speed v from kinetic energy — non-relativistic + relativistic, two-way solver.
- #1579 de Broglie wavelength = wavelength λ = h/p — computes v only as an intermediate step, not the main result.
- #1594 drift-velocity = speed drift electrons in a conductor (order of mm/s) — not the ballistic speed of free electrons.
- #1491 volts-to-ev = voltage → electron energy conversion — does not compute speed.
Frequently asked questions (FAQ)
1. What is the formula for electron speed from kinetic energy?
For non-relativistic electrons, kinetic energy E = 1/2 x m x v^2, so v = sqrt(2 x E / m), with m = 9.1093837015 x 10^-31 kg (electron mass, CODATA 2018). Energy must be in joules; 1 eV = 1.602176634 x 10^-19 J. Example: E = 100 eV = 1.602176634 x 10^-17 J gives v = 5.9309696 x 10^6 m/s.
2. What is the relativistic formula for electron speed?
For large kinetic energies, use the Lorentz factor gamma = 1 + E / (m_e x c^2), with m_e x c^2 = 510998.95 eV (electron rest energy). Speed v = c x sqrt(1 - 1/gamma^2), or beta = v/c = sqrt(1 - 1/gamma^2). Example: E = 1 MeV gives gamma = 2.956951 and v = 0.941079 c = 2.821285 x 10^8 m/s.
3. When must you use the relativistic formula?
Starting around E > 10 keV, or when v exceeds 20% of the speed of light (beta > 0.2). At 10 keV the classical formula already overestimates the speed by 1.46% (0.19499 c vs 0.19784 c). At 100 keV the error is 14.1%, and at 1 MeV the classical formula predicts v = 1.98 c — physically impossible. Below 1 keV the classical error is only 0.15% and still fine.
4. What is the speed of a 100 eV electron?
A 100 eV electron moves at v = 5.93010 x 10^6 m/s (relativistic), or 0.0197807 c (1.98% of the speed of light). The classical version gives 5.93097 x 10^6 m/s — a difference of only 0.0147%, so at 100 eV both formulas are practically identical. Lorentz factor gamma = 1.00019570.
5. What is the speed of a 1 MeV electron?
A 1 MeV electron moves at v = 2.821285 x 10^8 m/s = 0.941079 c (94.11% of the speed of light), with Lorentz factor gamma = 2.956951. The classical formula must not be used here: it predicts v = 1.978 c, faster than light — proof that the relativistic formula is required at MeV energies.
6. Can electrons move faster than light?
No. The relativistic formula guarantees v < c for any finite energy: even at E = 10 MeV, v = 0.9988176 c — still below the speed of light. If your classical formula yields v >= c (starting around E > 255.5 keV), that is a sign the formula is no longer valid and you must switch to the relativistic one.
7. How do you calculate kinetic energy from speed?
Classical: E = 1/2 x m x v^2. Relativistic: E = (gamma - 1) x m_e x c^2, with gamma = 1 / sqrt(1 - (v/c)^2). Example: v = 0.5 c (gamma = 1.154701) gives E = 79051.81 eV = 79.05 keV relativistically, while the classical formula gives only 63874.87 eV — a 23.8% difference. This calculator computes both versions at once.
8. What is the rest energy of the electron?
Electron rest energy m_e x c^2 = 510998.95 eV = 510.999 keV = 8.1871057768 x 10^-14 J. This value is the denominator in the Lorentz factor gamma = 1 + E_kinetic / (m_e x c^2), so a kinetic energy of 510998.95 eV exactly doubles the electron mass-energy (gamma = 2).
9. How is this calculator different from the de Broglie calculator #1579?
#1607 (this page) is the electron speed calculator: from kinetic energy directly to v, with both regimes (classical + relativistic) and a two-way solver. #1579 (de-broglie-wavelength) computes the de Broglie wavelength lambda = h/p — it computes v only as an intermediate step, not the main result. If you need the electron speed itself, use this page.
10. How is it different from electron drift velocity #1594?
Very different. #1594 drift-velocity computes the average drift velocity of electrons in a conductor (v_d = I / (n x A x q)) — on the order of mm/s, slow because of collisions. This #1607 computes the ballistic speed of a free electron from its kinetic energy — on the order of millions to hundreds of millions of m/s. Do not confuse them: drift velocity is not the speed of an individual electron.
11. Which energy units are supported?
eV (electronvolt), keV, MeV, and joules (J). 1 eV = 1.602176634 x 10^-19 J (exact, 2019 SI definition). Speed can be displayed in m/s, km/s, fractions of c (beta), or percent of the speed of light.
12. Why is my 10 keV result 19.50% of the speed of light?
Because that is the relativistically correct result: E = 10 keV gives gamma = 1.019569512 and beta = sqrt(1 - 1/gamma^2) = 0.1949856, i.e. 19.4986% of c. The classical formula gives 0.19784 c (19.78%) — overestimating by 1.46%. The 10 keV range is typical of medical X-ray tubes and SEM/TEM electron microscopes.
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