A sailor is in a sailboat on a river.

physics/sail Added material

Problem

A sailor is in a sailboat on a river.  The current is 3 knots with respect
to the land.  The wind (air velocity) is zero, with respect to the
land.  The sailor wants to proceed downriver as quickly as possible,
maximizing his downstream speed with respect to the land.

Should he raise the sail, or not?

Show solution

Depends on the sail.  If the boat is square-rigged, then not, since
raising the sail will simply increase the air resistance.

If the sailor has a fore-and-aft rig, then he should, since he can then
tack into the wind.  (Imagine the boat in still water with a 3-knot head
wind).

Additional material

[ Editor: a July 1994 posting, "FAQ Error? (physics/sail.p)", disputing
the solution above. ]

Okay, so here's a (fairly limited) description of the mechanics
involved:

Given a fixed frame of reference (the land) we can define some values:
  V: the Velocity of the boat, relative to the land.
  R: the (Real) velocity of the wind, relative to the land.
  E: the (Effective) velocity of the wind, relative to the boat.
  theta: the angle between V and E, (V.E)/(|V||E|)
  F(theta): the eFficiency of the sails for a given theta.
  P: the value of theta at which the sails start to pinch (excessive
     turbulence occurs behind the sails and power is lost).

N.B. The direction of the wind vectors point away from the oncoming
wind.  Also,
        E = R - V
        cos(theta) = V.E/|V||E|

The function F() looks something like the graph below.  (Please excuse
the poor ascii art).  This assumes that the sails are set correctly for
the angle theta.  The efficiency is greatest when the angle between the
E and V is zero (the effective wind is coming from over the stern of
the boat).  As the wind comes around towards the bow efficiency is lost
since the force from the bernoulli (sp?) effect is decreased and more
of the force from the wind is contributing to the sideways motion of
the boat causing increased friction on the underwater surfaces.  When
the sails start pinching (theta >= P) their effectiveness drops rapidly
until they provide more drag than forward force.

 F(theta)               theta = P
 |------______              |
 |            -----___      V
 |                    ---__
 |                         \
 |                          |
 |                          |
 |                          |
 0-------+-------+-------+--|----+-> theta
 |      pi/4    pi/2   3pi/4 \   pi
 |                            \__

So we have this dynamic system:
        V = E*F(theta)     (well, kind of!)

Returning to the original puzzle, R = 0, so E = -V, ie the effective
wind is always blowing over the bow of the boat.  Also, theta=pi and
the sails are causing negative forward force (drag).  Notice that this
is irrespective of the direction in which the boat is travelling, so
'tacking' achieves nothing.

There is a way to increase the strength of the effective wind called
'roll-tacking' which involves rolling the boat perpendicular to its
direction of motion and thus 'pushing' the sails through the air.

Now there does become a point when there is sufficient real wind (R)
such that an initial V will allow V to increase to beyond the modulus
of R.  The minimum R, V and the necessary value of theta depend on the
function F() and are probably reasonably difficult to calculate.

Can anybody offer some F()s for some real boats and the relationships
between the necessary values of R and V to maintain V?

-- Piers Haken <piersh@microsoft.com>

[ Editor: this analysis takes V relative to the land, but the keel
works against the water, which moves at 3 knots; relative to the water
the boat feels a 3-knot head wind, which is the point of the archive
solution. ]