Relativistic Energy Calculator
Calculate the total relativistic energy of a particle given its rest mass and velocity. Used in particle physics and nuclear engineering to account for energy at speeds where E = ½mv² is no longer accurate.
About this calculator
Einstein's famous mass-energy equivalence extends to moving objects. The total relativistic energy is E = γ m₀c² = (m₀c²) / √(1 − v²/c²), where m₀ is the rest mass of the object, c = 299,792,458 m/s is the speed of light, and γ is the Lorentz factor. When v = 0, this reduces to the rest energy E₀ = m₀c², the iconic equation of special relativity. The kinetic energy is then K = E − E₀ = (γ − 1) m₀c². At low velocities, K approximates the classical ½m₀v². At velocities near c, the total energy grows without bound because γ → ∞, which is why it is impossible to accelerate any mass to the speed of light — it would require infinite energy. This formula is foundational in nuclear physics, particle accelerators, and cosmology.
How to use
Find the total energy of a proton (m₀ = 1.6726 × 10⁻²⁷ kg) moving at v = 2.4 × 10⁸ m/s (80% of c). Step 1: Compute v²/c² ≈ 0.6401; √(1 − 0.6401) ≈ 0.5999. Step 2: E = (1.6726 × 10⁻²⁷ × (2.998 × 10⁸)²) / 0.5999. Step 3: m₀c² = 1.6726 × 10⁻²⁷ × 8.988 × 10¹⁶ ≈ 1.5033 × 10⁻¹⁰ J. Step 4: E = 1.5033 × 10⁻¹⁰ / 0.5999 ≈ 2.506 × 10⁻¹⁰ J, which is about 1.565 GeV — compared to the proton's rest energy of 938.3 MeV.
Frequently asked questions
What is the difference between rest energy and total relativistic energy?
Rest energy is E₀ = m₀c², the energy an object possesses simply by virtue of its mass, even when stationary. Total relativistic energy E = γ m₀c² includes both rest energy and kinetic energy. The kinetic contribution is K = (γ − 1) m₀c², which only matches the classical ½m₀v² at low speeds. At relativistic speeds, kinetic energy grows much faster than the classical formula predicts, making particle acceleration progressively more energy-expensive as you approach the speed of light.
Why can nothing with mass travel at the speed of light according to relativistic energy?
As a massive object accelerates toward c, the Lorentz factor γ = 1/√(1 − v²/c²) increases toward infinity. Since total energy E = γ m₀c², the energy required also approaches infinity. No finite energy source can supply this, making c an absolute speed limit for any object with rest mass m₀ > 0. Massless particles like photons travel exactly at c with finite energy because their rest mass is zero — they obey E = pc rather than E = γ m₀c².
How is relativistic energy used in particle accelerator physics?
In particle accelerators like CERN's LHC, protons are accelerated to γ values exceeding 7,000, meaning their total energy is over 7,000 times their rest energy. Engineers and physicists use the relativistic energy formula to calculate beam energies, collision kinematics, and the amount of energy available to create new particles. The famous relation E² = (pc)² + (m₀c²)² connects total energy, momentum, and rest mass and is used constantly in experimental high-energy physics to identify particles from detector measurements.