The title says it all.

If I was on a bus at 60 km/h, and I started walking on the bus at a steady pace of 5 km/h, then I'd technically be moving at 65 km/h, right?

So my son posed me an interesting question today: since light travels as fast as anything can go, what if I shined light when moving in a car?

How should I answer his question?


  If I was on a bus at 60 km/h, and I started walking on the bus at a
  steady pace of 5 km/h, then I'd technically be moving at 65 km/h,
  right?


Not exactly right.  You would be correct if the Galilean transformation correctly described the relationship between moving frames of reference but, it doesn't.

Instead, the empirical evidence is that the Lorentz transformation must be used and, by that transformation, your speed with respect to the ground would be slightly less than 65 km/h.  According to the Lorentz velocity addition formula, your speed with respect to the ground is given by:

$$\dfrac{60 + 5}{1 + \dfrac{60 \cdot 5}{c^2}} = \dfrac{65}{1 + 3.333 \cdot 10^{-15}} \text{km}/\text h \approx 64.9999999999998\ \text{km}/\text h$$

Sure, that's only very slightly less than 65 km/h but this is important to your main question because, when we calculate the speed of the light relative to the ground we get:

$$\dfrac{60 + c}{1 + \dfrac{60 \cdot c}{c^2}} = c$$

The speed of light, relative to the ground remains c!
