8.3650e-14
J
522,159.8002
eV
0.5222
MeV
0.03335557
1
939.157
MeV
8.3650e-14
J
522,159.8002
eV
0.5222
MeV
0.03335557
1
939.157
MeV
The Particle Energy Calculator computes the kinetic energy of a subatomic or nuclear particle from its mass and velocity, supporting both classical and relativistic calculations. This tool is essential for nuclear and particle physics, where particles routinely travel at significant fractions of the speed of light, making relativistic corrections crucial.
In the classical regime (\(v \ll c\)), kinetic energy is simply \(KE = \frac{1}{2}mv^2\). This approximation works well for thermal neutrons, alpha particles from radioactive decay, and ions in low-energy accelerators. However, for particles at even 10% of the speed of light, the classical formula underestimates the kinetic energy by about 0.75%, and the error grows rapidly at higher velocities.
In the relativistic regime, the correct kinetic energy is \(KE = (\gamma - 1)mc^2\), where \(\gamma = 1/\sqrt{1 - v^2/c^2}\) is the Lorentz factor. As \(v \to c\), \(\gamma \to \infty\) and the kinetic energy grows without bound—this is why no massive particle can reach the speed of light. For highly relativistic particles (like cosmic rays or LHC protons), the kinetic energy can exceed the rest mass energy by factors of thousands.
The calculator provides results in joules, electronvolts (eV), and megaelectronvolts (MeV), along with the relativistic parameters \(\beta = v/c\) and \(\gamma\). It also computes the total relativistic energy \(E = \gamma mc^2 = KE + mc^2\), which is conserved in all relativistic processes.
These calculations are fundamental for designing particle accelerators, analyzing nuclear reaction kinematics, computing radiation doses from charged particles, understanding cosmic ray interactions, and modeling particle detector responses. The energy of a particle determines its penetrating power, its ability to initiate nuclear reactions, and the type of Cherenkov or synchrotron radiation it produces.
Common particle masses: proton = \(1.673 \times 10^{-27}\) kg (938.3 MeV/c²), neutron = \(1.675 \times 10^{-27}\) kg, electron = \(9.109 \times 10^{-31}\) kg (0.511 MeV/c²), alpha particle = \(6.644 \times 10^{-27}\) kg.
Classical kinetic energy:
$$KE = \frac{1}{2}mv^2$$
Relativistic kinetic energy:
$$KE = (\gamma - 1)mc^2$$
where the Lorentz factor is:
$$\gamma = \frac{1}{\sqrt{1 - \beta^2}}, \quad \beta = \frac{v}{c}$$
Total relativistic energy:
$$E = \gamma mc^2 = KE + mc^2$$
Unit conversions:
$$E_{eV} = \frac{E_J}{1.602 \times 10^{-19}}, \quad E_{MeV} = \frac{E_J}{1.602 \times 10^{-13}}$$
For β < 0.1 (v < 30,000 km/s), classical and relativistic results agree to within 0.5%. For β > 0.5, always use relativistic. The Lorentz factor γ tells you the magnitude of relativistic effects: γ = 1 is non-relativistic, γ = 2 means KE equals the rest mass energy, and γ = 1000 (typical for LHC protons) means the particle has 1000 times its rest mass as kinetic energy.
Inputs
Results
A proton at 0.1c has KE = 4.7 MeV, while its rest mass energy is 938.3 MeV. The Lorentz factor of 1.005 means relativistic effects are only 0.5%—classical would work here.
Inputs
Results
At 0.99c, the proton's kinetic energy is 5,740 MeV—about 6 times its rest mass energy. The Lorentz factor of ~7 means the particle's total energy is 7 times its rest energy. Classical physics would be grossly incorrect here.
Use relativistic when v > 0.1c (about 3×10⁷ m/s). At 0.1c, the classical formula has ~0.5% error; at 0.5c, ~15% error; at 0.9c, classical is off by a factor of 3. In nuclear physics, always check β: alpha particles from decay (β ≈ 0.05) are safely classical, but electrons from beta decay (β often > 0.9) require relativity.
Total relativistic energy is E = γmc² = KE + mc², where mc² is the rest mass energy. Kinetic energy is only the energy due to motion: KE = (γ-1)mc². For a proton at rest, KE = 0 but E = 938.3 MeV. In particle collisions, total energy (including rest mass) is conserved, while kinetic energy may be converted to new particles.
As v → c, the Lorentz factor γ → ∞, meaning the kinetic energy diverges to infinity. To accelerate a massive particle to exactly c would require infinite energy. In practice, particles at major accelerators reach β = 0.999999991 (LHC protons at 6.5 TeV), tantalizingly close to but never reaching c.
Proton: 1.673 × 10⁻²⁷ kg | Neutron: 1.675 × 10⁻²⁷ kg | Electron: 9.109 × 10⁻³¹ kg | Alpha particle: 6.644 × 10⁻²⁷ kg | Muon: 1.884 × 10⁻²⁸ kg | Pion (π±): 2.488 × 10⁻²⁸ kg.
The energy of a charged particle determines its Linear Energy Transfer (LET)—the energy deposited per unit path length in matter. Higher energy particles penetrate deeper but may have lower LET. Alpha particles (high mass, moderate energy) have very high LET and cause intense local damage. The Bragg peak phenomenon, used in proton therapy, exploits the energy-dependent stopping power.
The proton rest mass energy is mc² = 938.272 MeV. This means creating a proton-antiproton pair requires at least 2 × 938.3 = 1876.5 MeV of available energy. The rest mass energy also sets the scale: when KE ≫ mc², the particle is "ultra-relativistic" and behaves almost like a massless particle.
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