Why would I waste my time? It has very little to do with a rocket!
You really can't think for yourself, can you?
Put your books down and think.
No, you wake up and do some serious thinking.
I do think and the projectile in the cannon or the "rocket" in the elevator shaft are pushed out by the force of the
compressed air of the back of the projectile or the "rocket".
On the other hand, a rocket in space is
pushed by the pressure of the burnt propellant on the
inside of the rocket nozzle.
Just put some effort into thinking about that and then explain what is wrong with it.
Rocket engines work better in a vacuum than in air - get used to the real world.
If you disagree please tell us at what air pressure the rocket engine ceases to provide thrust because the rockets keep accelerating way above 100,000 feet where no winged aircraft can fly - just how is that possible?
PS You can also explain the thrust by using action-reaction and get the same result.