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It is possible for a boat to use rudder to make a U-turn while coasting (moving by inertia), although it would lose some speed.

How exactly do boats trade a portion of magnitude of the initial linear momentum into a direction change? I would expect that the only force there is friction which would only reduce magnitude of the momentum, but it is actually changes its direction.

It must have to do with rudders, but I must admit I don't understand how they work.

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I recommend playing with a rigid, waterproof plate of some kind in a bathtub to get a better intuition of the forces involved. – user2357112 Aug 4 '14 at 13:03
up vote 5 down vote accepted

If you consider a boat moving through water, or alternatively a stationary boat, with water flowing past it, then the water has a momentum relative to the boat.

Water flowing past boat

The blue circle is supposed to show some small volume of the water of mass $m$. If the velocity of the water relative to the boat is $v$ then the momentum of this bit of water is $p = mv$.

Now look closely at the rear of the boat where the rudder is:


The water that hits the rudder is deflected so it moves in a different direction. Now momentum is a vector, so if the direction of flow changes the momentum of the water must change. The lower diagram shows this calculation. If we once again consider our small volume of water of mass $m$ it's original momentum is $mv$ and the new momentum is $p' = mv'$. The change in momentum is the vector $\Delta p$.

But Newton's laws tell us the the rate of change of momentum is just force:

$$ F= \frac{dp}{dt} $$

This means that changing the direction of the water produces a force in the direction of the vector $\Delta \vec{p}$, and this force is what pushes the rudder, and the rear of the boat, sideways and turns it.

If the rudder was perfect it would just change the direction of flow without changing the velocity of the water. In real life there is always turbulence in the lee of the rudder and this turbulence dissipates energy and creates a drag that slows the boat down.

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Thanks for the detailed explanation. I guess the piece I was missing is that at every moment the velocity of water relative to the rudder has the same direction. The note in the other answer about the keel absorbing sideways motion also helped me understand this (I plusoned it). – vitaly Aug 6 '14 at 5:42

There are two separate things at work here, the first is the rudder itself which changes the boat's facing. The second is actually the keel of the boat. This is what keeps boats moving in a straight line and is why a boat can do a "u-turn". The rudder turns the facing of the boat, because the keel makes it much easier for the boat to move forwards through the water rather than sideways this also causes the direction of movement to change as the boat rotates.

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a keel-less boat will turn just fine, tho' your basic point that the hull itself has to have some drag is accurate. – Carl Witthoft Aug 4 '14 at 13:14
@CarlWitthoft Without a keel the boat will turn but will continue moving in its original direction even while facing in a different direction. – Tim B Aug 4 '14 at 13:16
Tim, yes, which is why I put in the proviso that the hull have nonzero drag, essentially a "mini-keel." E.g., a Sunfish will turn with the daggerboard removed, but much less controllably. – Carl Witthoft Aug 4 '14 at 13:18

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