Stompy

I'm trying to wrap my mind around this and make sense of it:

An observer chasing a beam of light will measure it moving away from him at the same speed as a stationary observer.

Ok, so if someone was throwing a ball at you 20 feet per second and you run away 8 feet per second, the ball is then coming at you 12 feet per second. Therefore, if you ran 20 feet per second, the ball would appear motionless to you (briefly).

Let's say a beam of light is shining in a straight line along and infinite path, and you are traveling the speed of light on a parallel path right next to it, that light will still appear to be going the speed of light... how is this possible?

Maybe I'm asking an incredibly difficult question, but is there any "layman's terms" for why this is so?

[edit]

Maybe I asked this wrong... in other words: if you chase after a beam of light and you're going light speed, why isn't the beam of light motionless, how is it still going light speed from YOUR perspective?

Isn't the speed of light a constant? If so, once you reach that constant and your speed is matched with the beam of light you're chasing, how can it still appear to go faster? Wouldn't that NOT be constant then?

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