# Einstein's Formula for Adding Velocities

Einstein's Formula for Adding Velocities

According to the special theory of relativity, the composition of two collinear velocities and is given by a well-known formula derived by Einstein: , where is the speed of light, 2.9979× m/sec. For , this reduces to the simple Galilean formula . The result can be derived by successive application of collinear Lorentz boosts, but it can be shown more intuitively by an argument outlined in the details. In the Demonstration the speeds are scaled as (also known as ) to keep the two rocket ships within the graphic. Einstein's formula is analogous to the law of addition for hyperbolic tangents, , where corresponds to the rapidity, defined by .

u

v

u⊕v=

u+v

1+uv/c

2

c

10

8

u,v≪c

u⊕v=u+v

v/c

β

tanh(r+r)=

1

2

tanhr+tanhr

1

2

1+tanhrtanhr

1

2

r

β=tanhr