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Why we need big propellers at slow speeds

From Newton's laws we know that for every action there is an equal and opposite reaction. The push you get from a propeller is proportional to the momentum the propeller gives some water. We can think about this in the frame of reference of the original stationary water, and ignore the frame of reference of the boat.

The momentum is: mass * velocity

The energy put into this mass of water by the propeller is: 1/2 mass * velocity^2

Because the energy goes up with the square of the velocity and the momentum only with the velocity change, the most efficient approach is to push lots of water with a small velocity.

If a boat is going fast a propeller passes through lots of water and so can push a little bit on many different cubic feet of water and so only give any bit of water a small velocity.

A propeller that makes efficient use of 50 Hp at 15 MPH makes very inefficient use of the same power at 2 MPH. At slow speeds it goes through less water and gives it a much higher velocity.

An efficient propeller for a slow seastead will be large so that it can give a small velocity to a lot of water. Using a small fast propeller on a big slow seastead would take far more energy for the same push.

Power is energy/time. So the Hp or watts that we need will be higher if we do not design for an efficient use of the energy with a reasonable propeller. It would be easy for a bad design to use 10+ times more power than an optimal design.