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Relativistic Addition of Velocities

Main Concept

  In Newtonian   physics, if Alice is standing on a train moving at 100 km/h and throws a baseball forwards at 50 km/h, an observer Bob standing on the ground will see the ball travel 150 km/h. In the theory of special relativity however, addition of velocities is nonlinear:


u= v+u'1+vu'c2,


where   v is the speed of the moving observer relative to a 'stationary' observer, u' is the velocity of some projectile relative to the moving observer, u is the velocity of the projectile relative to the stationary observer, and c is the speed of light,   c=3×108m/s. 


Effects at low velocities


For low velocities, the effect of the difference between the relativistic formula and the classical formula is negligible. For instance, for a train traveling at 100 km/h = 27.8 m/s from which Alice throws a baseball forwards at 50 km/h = 13.9 m/s, the velocity seen by a stationary observer still works out to 150 km/h to a close approximation. In fact, it is 14 decimal places close, or less than one in 100 trillion! The ratio between the more correct relativistic addition and what you would get using the classical formula is:


                          u v + u' = 11+27.813.93×1082 =  0.9999999999999957.


So, the effects of special relativity are really not noticeable for slow moving objects here on Earth.


Effects at high velocities


On the other hand, the effects are very important when the velocities are high enough. Suppose an alien spaceship traveling at 0.9 c  towards Earth fires a missile at the planet with speed 0.2 c relative to the ship. According to the Newtonian addition of velocities, a stationary observer on Earth would see the missile traveling at 1.1 c, which is impossible, since it would arrive before the observer even saw it coming! Using the relativistic formula, however, we actually see the missile coming at:


                         u = 0.9 c+0.2 c1+0.90.2c2c2 = 0.93 c,


which is a much more reasonable speed, and would at least give us time to blink before being obliterated - assuming we were all glued to our telescopes when it happened.


The diagram below allows you to change your perspective, between the Earth, a spaceship, and a UFO. You can adjust the relative speeds of the ships with the sliders.

Reference Frame


Spaceship Speed Relative

to Earth( v/c ), v

UFO Speed Relative

to Spaceship ( v/c ), u'


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