A fly cast begins with a person raising the rod tip behind them while keeping tension in the line. Then the caster rockets their wrist forward. This flick of a wrist is all the momentum that is required to rocket the line forward. The small quick movement on the end of the rod translates to an incredibly fast traveling rod tip, acting like a lever arm spanning anywhere between 8 and 10 feet, resulting in a large amount of torque. Once the wrist has finished its flick around 1:30 position, then the momentum is transferred through the rod and to the rod tip which bends similar to a whip. The momentum doesn't stop there though, and this is the secret to a far reaching cast: the line. The momentum is transferred to the line which weighs considerably more than the fly itself in order to travel the needed distance. As the momentum travels from the fisherman’s arm, to rod, to line, the law of conservation of momentum applies. So, when momentum is transferred between each part the mass decreases, resulting in an increasing velocity of said part. By the time it reaches the end of the floating line to the leader, it is whippings through the air at incredible speeds. As mentioned before, this movement parallels the end of a whip.
This slowed down footage of a cast shows the momentum being transferred through all mediums in order to cast the fly.
So, while I can explain the physics behind a cast, I still can't explain why whenever I have the prime opportunity to cast towards a fish, my fly always ends up snagged in a tree. That though is a problem to be solved another day. I'm guessing it's some kind of undiscovered attractive force between the two.
As always thanks for reading! - ThePeculiarParticle