Velocity made good

Concept

Boats cannot sail directly into the wind, requiring the sailor to alternate between headings, which are commonly called "tacks." On a tack, the sailor will generally point the sailboat as close to the wind as possible while still keeping the winds blowing through the sails in a manner that provides aerodynamic lift to propel the boat. Then the sailor turns slightly away from the wind to create more forward wind pressure on the sails and better balance the boat, which allows it to move with greater speed, but less directly toward the mark.

Determining the velocity made good usually requires computation and instrumentation.

For example, in a northerly (wind coming from the north) on a heading of 60 degrees NE, the speed of the boat is 5.0 knots. Falling off to 65 degrees NE accelerates the boat to 5.2 knots. Turning up into the wind to a heading of 55 degrees NE causes the boat speed to drop to 4.0 knots. These data indicate the trade-off between speed and progress toward the upwind mark (to the north in this case). Finding the heading that moves the boat most quickly towards the mark requires basic trigonometry. The northward component of the boat's velocity vector is found by multiplying the boat speed by the cosine of the angle between the true wind direction (north) and the sailboat's heading.

In this case, the optimal VMG is obtained on a heading of 60 degrees off of the true wind (60 degrees NE or 300 degrees NW). Turning up into the wind (more towards the mark) makes less progress towards the mark because the boat slows down too much. Turning downwind speeds up the boat, but yields a course that leads too far away from the mark for the increased speed to be a benefit.

True VMG

The general concept of a boat's VMG as stated above is:

V M G , y ^ = V o c o s θ

Obviously this is a quick way to determine your VMG while sailing. The truth is though that the overview of the physics is much more complicated than that. Some instances to take into consideration is that a sailboats Speed Over Ground increases or decreases relative to the wind direction. In theory, a sailboats speed increases while sailing from upwind to a downwind direction, not before slowing down ( while officially going downwind) before sailing parallel to wind direction ( downwind). Some other factors to include is the sail boat specifics that exists for each manufacturer. This would include things such as 'Velocity Increase Constant', which is normally given by the manufacturer as a 'The sail boat increases speed by 10% for every X degrees from the wind'. Considering these factors in mind; a True VMG can be calculated with the equation:

V M G D , y ^ = V w c o s θ s ( 1 + β 100 ) | θ o − θ s | i c o s θ γ

The above equation is for finding the True VMG for a sail boat going UPWIND ONLY.

V M G D , y ^ = V w c o s θ s ( 1 + β 100 ) | 180 ∘ − θ s − θ o | i c o s θ γ

The above equation is for finding the True VMG for a sail boat going DOWNWIND ONLY.

Where:

V w = Velocity of the Wind

θ o = Angle between the direction of travel and the direction of the wind

θ γ = Angle between the direction of travel and the direction of destination

β = Velocity increase constant ( given by the manufacturer as a whole number ( ex: 10% per x degrees; β = 10 ))

θ s = No Go Zone Limit ( given by manufacturer )

i = Degree Interval ( given by manufacturer; ex: (x% per 5 ∘  ; i = 5 ))

Even though the above equations isn't the most ideal method to use while you are on the water, but it gives knowledge of a more precise theoretical number to true VMG. To find the True VMG, all one would need to do is measure three variables: wind velocity, angle between the direction of travel and direction of the wind, and angle between direction of travel and direction of destination. V w , θ o , and θ γ .

VMG Rate of Change

Taking the two equations for True VMG given the section above, calculating the rate of change in respect to the angle towards your destination is achieved by differential calculus. Simply taking the derivative of each equation will allow you to calculate the rate of VMG per angle away from your destination. This is another tool to help prove the concept of VMG.

The equation to calculate True VMG in the upwind direction:

V M G D , y ^ = V w c o s θ s ( 1 + β 100 ) | θ o − θ s | i c o s θ γ

To find the rate of change for VMG in respect to your angle towards destination:

f ′ ( θ γ ) = d V M G U d θ γ = − V w c o s θ s ( 1 + β 100 ) | θ o − θ s | i s i n θ γ

Example:

V w = 10 knots

θ o = 60

θ γ = 40

β = 10

θ s = 40

i = 5

V M G D , y ^ = 14.6 knots towards destination.

f ′ ( θ γ ) = d V M G U d θ γ = -12.3 knots per degree away from destination.

*** Table uses same values as above example.