Particle Velocity Calculator

Calculator

Formula used

How to use this calculator

1. Select a calculation mode that matches your known quantities.
2. Choose a model. Use relativistic for high-energy particles.
3. Enter values with units. Use scientific notation when needed.
4. Press Calculate to view velocity, beta, and gamma above.
5. Use CSV or PDF buttons to export your computed result.

Professional article

1) What particle velocity represents

Particle velocity is the rate of position change, reported here in m/s and km/s. In experiments it links directly to time-of-flight, beam transport, diffusion, and collision rates. This calculator also reports β = v/c and the Lorentz factor γ when speeds become significant.

2) Classical versus relativistic regimes

At low speeds, classical formulas such as v = d/t, v = p/m, and v = √(2K/m) are accurate. As v approaches the light speed c = 299,792,458 m/s, relativistic relations prevent unphysical values and correctly capture time dilation through γ.

3) Momentum-based estimates in practice

Momentum is often measured or inferred from magnetic rigidity and tracking. Classically, v = p/m is immediate, but relativistically the same momentum can correspond to a higher total energy. Using v = p c² / √[(p c)² + (m c²)²] keeps the result consistent with energy–momentum physics.

4) Energy-based velocity for beams

Kinetic energy is common in accelerator and plasma diagnostics. For an electron, the rest energy is about 511 keV, so even 100 keV is partially relativistic. The calculator converts energy units (eV to GeV) to joules internally and then returns v, β, and γ for interpretation.

5) Total energy and the Lorentz factor

When total energy E is known, γ = E/(m c²) directly determines β through β = √(1 − 1/γ²). This mode is useful for high-energy particles where rest energy is a small fraction of E, and small changes in γ can produce large changes in β.

6) de Broglie wavelength as a velocity route

Wave–particle duality provides p = h/λ with Planck’s constant h = 6.62607015×10⁻³⁴ J·s. Combining this momentum with a known mass yields velocity. This pathway is helpful in electron microscopy and diffraction, where λ is measured or specified by instrumentation.

7) Unit handling and numerical stability

Reliable velocity estimates depend on consistent units. The calculator supports common distance, time, mass, energy, momentum, and wavelength units and converts them to SI before computing. If a classical configuration suggests v ≥ c, a warning appears so you can switch to a relativistic model.

8) Typical values and interpretation

Macroscopic motion may be meters per second, while space plasmas can reach hundreds of km/s. Laboratory electron beams commonly span tens to hundreds of keV, producing β values well above 0.5. Use γ to quantify how strongly relativistic effects influence timing and energy diagnostics.

FAQs

1) When should I prefer the relativistic model?

Use the relativistic model when your result is a noticeable fraction of light speed, or when energies are comparable to rest energy. It enforces v < c and provides meaningful β and γ for interpretation.

2) Why can classical formulas give v greater than c?

Classical relations ignore the energy–momentum constraints of special relativity. At high momentum or energy, they can overestimate speed. Relativistic formulas couple momentum, energy, and mass so the speed never exceeds c.

3) What is β and why is it useful?

β is the ratio v/c, a convenient dimensionless speed measure. It helps compare particles across unit systems and quickly indicates whether relativistic effects are important. Many radiation and timing formulas are naturally expressed in β.

4) What does γ represent physically?

γ is the Lorentz factor that quantifies relativistic time dilation and energy scaling. It equals 1 at low speeds and grows as v approaches c. In energy terms, kinetic energy relates to (γ − 1) m c².

5) Can I compute velocity from wavelength for any particle?

Yes, if you know the particle mass and a de Broglie wavelength. The calculator uses p = h/λ, then converts momentum to speed using either classical or relativistic relations, depending on your model selection.

6) How accurate are the example table values?

They are illustrative, rounded values meant to demonstrate workflows. Your exact result depends on chosen units, significant figures, and whether you select classical or relativistic modeling. Always verify inputs match your experimental definitions.

7) What should I export: CSV or PDF?

Export CSV if you want to log results, plot trends, or import into spreadsheets. Export PDF when you need a clean, shareable snapshot for a lab notebook, report, or quick review meeting.