Relativistic Velocity & Momentum
Relativistic velocity addition, Lorentz factor, momentum, kinetic energy, and Doppler shift.
Velocity addition: w = (u + v) / (1 + uv/c²). Two objects each moving at 0.9c relative to you combine to less than c — never exceeding the speed of light.
e.g. 0.9 = 90% of c
Frame velocity relative to observer
Combined velocity w = (v₁+v₂)/(1+v₁v₂/c²)99.447514% c298136.1 km/s
Classical (wrong): v₁+v₂180.0000% c
Relativistic result w99.447514% c
γ for v₁2.2942
γ for v₂2.2942
γ for w9.5263
Momentum at v₁618994000 kg·m/s
Kinetic energy at v₁1.1631e+17 J
Momentum at w2840139000 kg·m/s
Kinetic energy at w7.6631e+17 J
Relativistic Doppler shift (source at v₁ toward observer)
Frequency ratio f_obs/f_source4.358899
Redshift z (recession)-0.770584
Blueshift factor (approach)4.358899×
Special relativity
Velocities must be given as a fraction of c (e.g. 0.9 = 90% of c). Negative values represent motion in the opposite direction. The combined velocity always stays below c regardless of how close v₁ and v₂ are to c.