NSXLNT said:
It would seem that the missile is already traveling at the speed of sound just being carried by the plane, and then, when fired, it would reach it's max of mach one under it's own power, thus doubling it's actual speed?
No. Let's say you have a bicycle. At maximum effort you can pedal it 20 MPH on a flat road. Now tie the bicycle to the rear bumper of a car with a rope. Get on the bike and have someone accelerate the car up to 30 MPH on a flat road. Start pedaling as fast as you can and take a knife and cut the rope.
What happens? Do you gradually:
A) Accelerate to 50 MPH (your max speed + the speed of the car)
B) Decelerate to 20 MPH (your max speed under your own power)
Obviously the answer is B. The same is true for the self-propeled air-to-air missile fired from the plane.
The reason air-to-air missiles can catch planes in real life is because they can fly faster under their own power than the planes. For example an AIM-7 Sparrow runs at Mach 4 under it's own power -- it would reach Mach 4 even if you launched it from a tower. A missile that couldn't go considerably faster than the target plane under it's own power would be useless.
So it's more like being towed up to 5 MPH by the car, then you cut the rope and accelerate under your own power up to 20 MPH... It's the same speed you were able to reach on your own, though you get there a little faster since you get a head start by being towed up to 5 MPH.
But again this has no bearing on the way light behaves. The original question does not really have an answer because it there are no known physics decribing the situation of an object with mass travelling at the speed of light. It's like asking "What is the result if I divide a number by zero?" You can't, so there isn't an answer.
You can try to describe what should happen if you were travelling very close to the speed of light and turned on your headlights, as that situation involves a scenario that is believed to be possible. If anyone is really interesting in learning more, make a big bowl of popcorn and do a bunch of reading here:
http://www.weburbia.demon.co.uk/physics/relativity.html