Relativistic Momentum and Energy
Relativistic momentum and energy preserve conservation laws in every inertial frame. A particle's rest mass determines its rest energy, while its total energy also depends on its motion.
What a learner can do afterwards
- Computes relativistic momentum, kinetic energy and total energy for a fast particle
- Relates energy, momentum and rest mass and applies the relation to a massless particle
- Explains rest energy and where it shows up in nuclear reactions
1 · Read
Let m be rest mass, u speed and c vacuum light speed. In classical mechanics, momentum is m u and kinetic energy is one half m u squared. Momentum p = gamma m u. Gamma = 1 / sqrt(1 - (u/c) squared) increases with speed and approaches 1 near rest.
Total energy E = gamma m c squared. At rest gamma = 1, giving rest energy m c squared even without motion.
Fission, fusion and annihilation can release rest energy. Decreased total particle rest mass supplies kinetic energy or radiation, conserving total energy.
E squared = (pc) squared + (m c squared) squared, where p is momentum magnitude. For photons m = 0, so E = pc and p = E/c. Photons travel at c in vacuum.
Compute gamma = 1 / sqrt(1 - (u/c) squared), then p = gamma m u, E = gamma m c squared and K = (gamma minus 1) m c squared. At u = 0.6c, gamma = 1 / sqrt(0.64) = 1.25, giving K = 0.25mc squared.
Released rest energy equals the total rest-mass decrease times c squared. Square c before multiplying. The unit kg m squared / s squared equals J.
As u approaches c, gamma, momentum and energy grow without bound. No finite energy can accelerate a massive particle to c.
Stretch momentum and energy by gamma, keep rest energy in mass, and let massless light obey E equals p c.
Original lesson and quiz text authored for this learning dataset.
Reading summaries adapted from OpenStax's College Physics, sections 28.5 and 28.6, and University Physics Volume 3, section 5.9.
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