Vector addition in speeds close to that of light I've been studying special relativity and found a problem (at least for my limited knowledge): There's a car inside a train that is moving at speed $V_t$, for $V_t = 0.5c$. Once the car accelerates with $V_c = 0.5c$, from a reference frame other than those of the car and the train, does $V_t + V_c = c$?
If I am travelling on a car at around 60 km/h, and I shine a light, does that mean that the light is travelling faster than the speed of light? 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?
How to derive $E=mc^{2}$? Is there some way to derive ? I can understand that energy in something is proportional to its mass, but the $c^{2}$ part. I have no idea. It seems like the way the units are going it would end up as $kg*(m/s)^{2}$, which I think is the unit for Newtons, not energy. And why is there no constant of proportionality $k$? Did Einstein just set it up perfectly so that there wouldn't be one?