Physics for beginners/10-a puzzling inconsistency

File:Physics for beginners-10-a puzzling inconstistency.pdf

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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 ($10^6$) times faster than sound.

Mathematically, this model for relative motion is the equation:

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

where:

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

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

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

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