momentum equation control volume
A jet of water hits a plate. When the jet strikes the plate and turns, where does the force on the plate come from? Drag the strength of the jet and feel it. The harder you stand the water up, the harder the plate is pushed, and that push is the place where momentum changed.
The jet hits the plate and scatters sideways, and when you drag the strength of the jet the force on the plate grows. By as much as the plate stopped the momentum of the water, the plate is pushed by the reaction. Shoot hard and you are pushed back hard. Shoot softly and you are pushed back softly.
Bundle it in a control volume and look at the momentum coming in and going out, and the structure shows. The difference between the incoming momentum arrow and the outgoing momentum arrow is exactly the force applied to the flow. Momentum does not vanish. Only the direction changes, so that change appears as a force. Compare what entered and what left in the stretch the box wraps, and a force remains.
Drag the turning angle and the more you turn the flow the larger the change of momentum and the larger the reaction. Turn it fully back through 180 degrees and the change is the largest, so the force doubles. Glance it aside and the force is small. Turn it back on itself and the force is the largest.
In symbols, F = ṁ Δv, the mass flow rate times the change of speed. The mass flow rate is ṁ = ρAv, so F = ρAv·Δv, and when you drag v the v grows in two places and the force grows as v squared. Speed and amount grow the force together.
Look at a fire hose and by as much as the hose shoots a lot of water out fast, the reaction pushes the firefighter backward. Drag the flow rate up and the reaction jumps, and a rocket moves forward by the same principle, blowing gas backward. The momentum sent backward is the force that pushes forward.