Relativistic Velocity Addition Calculator
Enter observer speed v, object speed u' and light speed c to compute u=(v+u')/(1+vu'/c²). v=0.5c, u'=0.5c → u≈0.8c (not 1.0c); v=c, u'=c → u=c (light-speed limit).
Input Data
Results
At a glance:Relativistic velocity addition (Albert Einstein 1905, 'On the Electrodynamics of Moving Bodies'): in special relativity, when an observer moves at speed v and an object moves at u' in the observer's frame, its speed in the rest frame is u=(v+u')/(1+vu'/c²), c the speed of light. Physical meaning: (1) when v, u'≪c the denominator≈1, u≈v+u' (Newtonian approximation); (2) when v or u' approaches c the denominator exceeds 1, and the sum is less than v+u'; (3) when v=c and u'=c, u=(c+c)/(1+1)=c — the speed of light is unbeatable, and two c's still add to c; (4) any sub-c sum stays sub-c. Derivation: from the Lorentz transform x'=γ(x−vt), t'=γ(t−vx/c²), differentiating gives dx'/dt'=u', dx/dt=u, and u=dx/dt=(dx'+v dt')/(dt'+v dx'/c²)=(u'+v)/(1+vu'/c²). History: Einstein's 1905 special relativity overthrew the ether assumption, with the constancy of light speed as its cornerstone. Applications: (1) particle accelerators (LHC protons at 0.999999991c); (2) space travel (relativistic effects significant); (3) relativistic dynamics, momentum and energy; (4) GPS satellite speed correction; (5) muon lifetime extension (cosmic rays).
Formula
Velocity addition: u = (v + u') / (1 + v·u'/c²)
Newtonian limit: v, u' ≪ c → u ≈ v + u'
Light-speed limit: v=c, u'=c → u = c
Lorentz transform: x' = γ(x − vt), t' = γ(t − vx/c²)
Lorentz factor: γ = 1/√(1 − v²/c²)
$$u = \frac{v + u'}{1 + \frac{v u'}{c^2}}, \quad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}, \quad x' = \gamma(x - vt)$$How to Use
- Enter observer speed v (m/s, 0.5c=1.5e8).
- Enter object speed u' (m/s, 0.5c=1.5e8).
- Enter light speed c (m/s, 299792458).
- The tool computes the resultant u and ratio u/c.
Case Studies
Particle accelerator and proton collisions
CERN LHC protons at 7 TeV move at 0.999999991c. Two beams collide head-on; the sum u=(0.999999991c+0.999999991c)/(1+0.999999991²)≈0.999999991c (not 1.999999982c).
But the centre-of-mass collision energy is 14 TeV (two 7 TeV beams). Relativistic momentum p=γmv, energy E=γmc², not velocity addition.
HKU physics participates in LHC ATLAS, hunting the Higgs (2012, Nobel 2013). Relativistic addition underpins particle physics.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.