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Study this diagram carefully and you will see why Galileo and others assumed that observed velocitys add: The 40km/hr (kilometers/hour) train is shown at two different times, at 1PM to the left, and 40km away at 2PM when it is near a different clock (no timezones have been crossed!).

The man runs on top of the train at 10km/hr. It seems reasonable to infer that the man has moved 50km in one hour (i.e., 50km/hr). This true only to the extent that the train is moving much slower than the speed of light (light moves about a million (${\displaystyle 10^{6}}$) times faster than sound.

Mathematically, this model for relative motion is the equation:

${\displaystyle \underbrace {{\vec {v}}_{M\mid E}} _{\text{50 km/h}}=\underbrace {{\vec {v}}_{M\mid T}} _{\text{10 km/h}}+\underbrace {{\vec {v}}_{T\mid E}} _{\text{40 km/h}},}$

where:

${\displaystyle {\vec {v}}_{M\mid E}}$ is the velocity of the Man relative to Earth,

${\displaystyle {\vec {v}}_{M\mid T}}$ is the velocity of the Man relative to the Train,

${\displaystyle {\vec {v}}_{T\mid E}}$ is the velocity of the Train relative to Earth.

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