system
Given the following post, respond as if you're commenting on Reddit:
human
Rocketship going 51%c shoots out a second rocket ship going 51%c?

 I sometimes help tutor remedial introductory uni-physics, and someone asked this question and it honestly stumped me I couldn’t figure out an intuitive way to answer this basic question.

The student asked something like: “If you had a rocket ship that was travelling through space at 51% of the speed of light relative to earth and it had a smaller rocket ship on board and it shot off the second spaceship from its docking bay and the smaller rocket ship also accelerated itself to 51% the speed of light relative to the first ship. Wouldn’t the smaller rocket ship be travelling at 102% the speed of light relative to earth?”

I started talking about frames of reference but honestly, I can’t for the life of me figure out an intuitive way of explaining that. Pleas help.
gpt
This is the core of special relativity. Galilean relativity works exactly like your student expects it to, and it's a fine approximation to the world in everyday use, but fails in extreme circumstances. 

An intuitive way to think about this is that distance (and therefore any kind of velocity, which is just a distance per time) is not *really* expressed by just spatial dimensions. There's a 4th component, which is the time coordinate, and it acts very differently from the spatial dimensions. Importantly, distance is not:

    ds^2 = dt^2 + dx^2 + dy^2 + dz^2 

But instead the time component is *negative*:

    ds^2 = - dt^2 + dx^2 + dy^2 + dz^2

This means that moving between reference frames happens with a different kind of transformation than the ones you're used to, it happens with a **Lorentz transformation**. (A book to read a bit more about this is Feynman's "6 not-so-easy pieces" which explores the new geometry in a reasonably approachable, but not equation free, way.)  

That's it, really. Distance (and therefore speed) includes this weird, negative, time component that changes how the measurement is made. Physical distance is *not* conserved when things speed up or slow down. This other, strange-to-our-brains distance **is**.