This is for a class on special relativity I am to give to some school children.
Moe is moving at 0.9c. At the instant he passed Joe he emits a flash of light. One microsecond later, Joe(at rest) sees that Moe has traveled 270m and the light flash has traveled 300m. That is, the distance between Moe and the light flash is 30m. All perfectly good so far.
But Moe, who does not know he is moving, waits one microsecond and observes that the light has traveled 300m in front of him. Also perfectly good.
The reason that Moe sees that the light has traveled 300m and not 30m is that his time has been dilated, so that Moe's microsecond is longer than Joe's microsecond. Also, when he measures how far ahead the light flash is, he is using a shortened ruler.
All very good. Except that if you use the Lorenz transformation equations, you find that his time has dilated to 2.29 microseconds and his ruler is shorter by 1/2.29. Those are not enough to explain a 300m measurement for Moe: It only provides for the light to be 157m ahead, not 300m ahead!
What am I missing here?