How this instrument works
The formula rearranges the definition of kinetic energy, KE = ½mv², to solve for speed instead of energy: v = √(2·KE ⁄ m). Because energy scales with the square of velocity, the relationship is not linear — doubling a particle's kinetic energy raises its velocity by only about 41 percent, not 100 percent, and quadrupling the energy is what actually doubles the speed. That square root is why heavy particles carrying large energies can still be comparatively slow, and light particles carrying tiny energies can already be moving fast.
The ½mv² term traces back to the vis viva, or 'living force,' that Gottfried Leibniz proposed in the 1680s as the quantity conserved in collisions, a rival to Descartes' simpler mv. Gaspard-Gustave de Coriolis supplied the modern factor of one-half and the name 'kinetic energy' in 1829, and the algebra used here has not changed since. The formula is purely classical: it assumes Newtonian mechanics, where momentum and energy behave like ordinary numbers rather than the stretched quantities relativity describes.
That assumption fails once a particle's speed becomes an appreciable fraction of the speed of light. An electron accelerated through a few hundred volts still obeys this formula comfortably, but push the accelerating voltage into the tens of kilovolts — the range used in older CRT tubes and electron microscopes — and the true, relativistic speed falls noticeably below what v = √(2·KE ⁄ m) predicts, because some of the added energy goes into relativistic mass increase rather than speed. For everyday thermal- and lab-scale energies, though, the classical result holds to many decimal places.
- Enter Kinetic energy in joules — single particles carry tiny values, so use scientific notation like 1e-19.
- Enter Particle mass in kilograms; it defaults to the electron mass, 9.109×10⁻³¹ kg, but any particle's rest mass works.
- Read Particle velocity in metres per second, computed as the square root of twice the energy-to-mass ratio.
- Change the mass to compare particles: swapping in a proton's mass at the same energy shows how much slower a heavier particle moves.
Worked example — an electron with 1×10⁻¹⁹ J of kinetic energy
Feed the defaults into the instrument: Kinetic energy = 1×10⁻¹⁹ J and Particle mass = 9.109×10⁻³¹ kg, the rest mass of an electron. That energy is about 0.624 eV, comparable to the thermal energy an electron might pick up from ordinary collisions at room temperature, nothing exotic. The formula returns Particle velocity = 468,575.575 m/s, a little over 468 kilometres per second.
That speed is only about 0.156 percent of light speed, so treating it classically is entirely safe here, but the figure makes a useful benchmark. It shows how casually electrons reach speeds that would be extraordinary for anything with mass, simply because an electron's mass is so small that even a modest 1×10⁻¹⁹ J produces a large square root. Swap in a proton's mass, 1.673×10⁻²⁷ kg, about 1,836 times heavier, for the same energy, and velocity drops to roughly 10,934 m/s — over 40 times slower, because velocity shrinks only with the square root of that mass ratio, not the full 1,836 times.
Questions
Why does doubling the kinetic energy not double the velocity?
Because kinetic energy grows with the square of speed, KE = ½mv², the inverse relationship carries a square root: v = √(2·KE ⁄ m). Doubling KE multiplies velocity by √2, about 1.41, not by 2. To actually double a particle's speed you must quadruple its kinetic energy, the same square-law behaviour that governs braking distances and wind loads.
Is this formula still accurate for fast-moving electrons?
Only up to a point. v = √(2·KE ⁄ m) is the classical, Newtonian result, and it starts to overestimate the true speed once velocity approaches a meaningful fraction of light speed — noticeably by the tens-of-kilovolts range used in electron microscopes and older CRT tubes. Past that, a relativistic energy-speed relation is needed instead of this one.
What mass should I use for a proton or another particle?
Enter the rest mass of whichever particle you're tracking: 1.673×10⁻²⁷ kg for a proton, 1.675×10⁻²⁷ kg for a neutron, or a specific ion's mass in kilograms. The default, 9.109×10⁻³¹ kg, is the electron; swapping it out is the only change needed, since the formula and its algebra stay identical.
Why is the kinetic energy field usually such a tiny number?
Because single particles carry vanishingly small energies compared to everyday objects — a joule is roughly what it takes to lift an apple a few centimetres, while one electron typically carries somewhere between 10⁻²⁰ and 10⁻¹⁷ joules. Enter the value in scientific notation, such as 1e-19; no unit conversion is needed since the field already expects joules.
Who actually calculates particle velocity this way?
Semiconductor engineers running ion implanters use it to convert an accelerating voltage and known ion mass into an implant velocity and penetration depth; mass-spectrometry technicians use the same algebra to interpret time-of-flight data; and physics students use it to check that a given energy corresponds to a plausible, non-relativistic speed.
Does the particle's electric charge matter for this calculation?
Not directly. Once you know the kinetic energy in joules, charge has already done its job — if that energy came from accelerating the particle through a voltage V, the charge determined how much energy it picked up via KE = qV, but v = √(2·KE ⁄ m) only needs the resulting energy and the mass, regardless of what produced the energy.