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Einstein's Formula for Adding Velocities

​
mother ship speed u/c
0.5
satellite launch speed v/c
0.5
time t
0
According to the special theory of relativity, the composition of two collinear velocities
u
and
v
is given by a well-known formula derived by Einstein:
u⊕v=
u+v
1+uv/
2
c
, where
c
is the speed of light, 2.9979×
8
10
m/sec. For
u,v≪c
, this reduces to the simple Galilean formula
u⊕v=u+v
. 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
v/c
(also known as
β
) to keep the two rocket ships within the graphic. Einstein's formula is analogous to the law of addition for hyperbolic tangents,
tanh(
r
1
+
r
2
)=
tanh
r
1
+tanh
r
2
1+tanh
r
1
tanh
r
2
, where
r
corresponds to the rapidity, defined by
β=tanhr
.
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