At speeds much smaller than the speed of light, Newtonian formulas like p = mv and KE = 1/2 mv^2 work very well. Near the speed of light, these formulas fail because space and time are linked by special relativity. Relativistic energy and momentum explain why massive objects can get closer and closer to light speed but never reach it.
This idea matters in particle accelerators, cosmic rays, nuclear physics, and high energy astronomy.
The key factor is the Lorentz factor, γ = 1 / sqrt(1 - v^2/c^2), which grows rapidly as speed v approaches the speed of light c. Relativistic momentum is p = γmv, total energy is E = γmc^2, and rest energy is E0 = mc^2. The kinetic energy is the extra energy above rest energy, so K = (γ - 1)mc^2.
Energy and momentum are connected by E^2 = (pc)^2 + (mc^2)^2, which also works for massless particles when m = 0.
Key Facts
- Lorentz factor: γ = 1 / sqrt(1 - v^2/c^2).
- Relativistic momentum: p = γmv.
- Total relativistic energy: E = γmc^2.
- Rest energy: E0 = mc^2.
- Relativistic kinetic energy: K = E - E0 = (γ - 1)mc^2.
- Energy-momentum relation: E^2 = (pc)^2 + (mc^2)^2.
Vocabulary
- Lorentz factor
- The factor γ that measures how strongly time, length, energy, and momentum change at relativistic speeds.
- Rest energy
- The energy an object has because of its mass even when it is not moving, given by E0 = mc^2.
- Total energy
- The full relativistic energy of a particle, including both rest energy and kinetic energy.
- Relativistic momentum
- The momentum of an object moving at any speed, calculated with p = γmv instead of just p = mv.
- Energy-momentum relation
- The equation E^2 = (pc)^2 + (mc^2)^2 that connects a particle's total energy, momentum, and rest mass.
Common Mistakes to Avoid
- Using p = mv at speeds near c. This is wrong because momentum must include the Lorentz factor, so the correct formula is p = γmv.
- Treating kinetic energy as 1/2 mv^2 for relativistic particles. This underestimates the energy at high speed because the correct expression is K = (γ - 1)mc^2.
- Forgetting that total energy includes rest energy. Total energy is E = γmc^2, while kinetic energy is only the part above mc^2.
- Assuming a massive object can reach the speed of light if enough energy is added. This is wrong because γ grows without limit as v approaches c, so reaching c would require infinite energy.
Practice Questions
- 1 A proton has rest energy 938 MeV and moves at 0.80c. Calculate γ, its total energy, and its kinetic energy.
- 2 An electron has rest energy 0.511 MeV and momentum p = 2.00 MeV/c. Use E^2 = (pc)^2 + (mc^2)^2 to find its total energy.
- 3 Explain why adding the same amount of energy to a spacecraft near light speed produces a smaller increase in speed than it would at low speed.