The object of this little exercise is to show 'something' about time. I was going to follow up with spacetime intervals where time is very obviously a dimension but I drifted outside my comfort zone. I suspect what follows will cause enough controversy for the immediate future.
We need a flagpole by the side of the road. If there isn't one we have to find one and put it there. The flagpole needs to be vertical. The road needs to be horizontal. If the road isn't horizontal and the flagpole isn't vertical then Pythagoras won't be able to do his thing so we need to get that right to start with.
We have the lovely Alice on the beach - we'll say she is in the 'beach frame'. We have the microbus which we'll refer to as the 'microbus frame' because it can move in the beach frame. Crazy Carlos drives the microbus and Bob sits in the back watching what happens. Alice has some of the light show stuff from the microbus and is sitting at the bottom of the vertical pole with it. Coupla beers from the osokool and we're ready to go.
Here is what is going to happen. Crazy Carlos is going to drive the Microbus past the flagpole and as it passes Alice is going to send a single pulse of light up the flagpole from point A so it hits a thing (say) 5 metres up at point B. We'll call this vertical distance travelled by the pulse y.
Here we go. Microbus thunders towards the flagpole. As it passes there's the pulse going up the flagpole. Bob sees that the flagpole is moving backwards as he passes so for him when the pulse hits the top the flagpole it isn't at a point vertically above the bottom its at a point we'll call C.
The distance the flagpole travels in the microbus frame (edited here) is going to be 'how long the pulse takes to travel in the microbus frame' (call this t) times the speed of the microbus (call that v).
So in your triangle (you are drawing one aren't you?) you have x and y at right angles where y is the height of the pole and x is the distance the microbus moved while the pulse travelled. Using Pythag we see the distance the pulse actually travelled in the microbus frame (call it z) is given by z²=x²+y² (1).
Knowing the speed of light is always c...
Using T for the time in the beach frame we have:
y=cT
Using t for the time in the microbus frame we have:
x=vt and
t=z/c or z=ct
substituting x,y and z into (1)
c²t² = v²t² + c²T²
or
c²T² = c²t²- v²t²
or
c²T² = t²(c²- v²)
or
T² = t²(1- v²/c²)
or
T = t √(1- v²/c²)
Since (1- v²/c²) < 1 when v > 0 Bob concludes the elapsed time on the beach is less than the elapsed time in the microbus. Note that Bob is stationary in the microbus frame and he sees the flagpole and beach frame moving past at velocity v.
You missed the whole point .
We ALL know that we measure time , WE ALL get this .
What you missed ; is that without the MOVEMENT of the bus , there is nothing to measure .
But more importantly , time did not move the bus .