# Addition of velocities

A velocity-addition formula is a 3-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. The speed of light in the fluid is slower than the speed of light in vacuum, and it changes if the fluid is moving along with the light. In 1851, Fizeau measured the speed of light in a fluid moving parallel to the light using a Michelson interferometer. Fizeau's results were not in accord with the then prevalent theories. Fizeau experimentally correctly determined the zeroth term of an expansion of the relativistically correct addition law in terms of V/c .

### Relativistic velocity addition formula

**u = (v+w) /(1 + (v * w)/c²)**

where:

u is speed of the projectile when outside of the spaceship

v is the speed of the spaceship

w is the speed of the projectile when we see from the spaceship

c is the speed of light

The Einstein velocity relationship transforms a measured velocity as seen in one inertial frame of reference (u) to the velocity as measured in a frame moving at velocity v with respect to it (u'). In problems involving more than two objects, the main difficulty is the assignment of velocity to all the objects.

### Standard applications of velocity-addition formulas

1. Aberration of light

2. Doppler shift

3. Doppler navigation

Velocity is a vector quantity, so the relationships between the magnitude, direction, x-axis component and y-axis component are important.

These vector components can be added analytically or graphically.

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