Physics for beginners/10-a puzzling inconsistency

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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 ( $1 0^{6}$ ) times faster than sound.

Mathematically, this model for relative motion is the equation:

\underset{50 km/h}{\underset{⏟}{{\vec{v}}_{M ∣ E}}} = \underset{10 km/h}{\underset{⏟}{{\vec{v}}_{M ∣ T}}} + \underset{40 km/h}{\underset{⏟}{{\vec{v}}_{T ∣ E}}},

where:

{\vec{v}}_{M ∣ E}

is the velocity of the Man relative to Earth,

{\vec{v}}_{M ∣ T}

is the velocity of the Man relative to the Train,

{\vec{v}}_{T ∣ E}

is the velocity of the Train relative to Earth.

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