Sailboat Speed Question: Can a Motor Increase Speed Beyond 10 Knots with Sails?

[DaveC426913]

Your specs sound like a MacGregor 26;

I guess you can tell I like talking about boats.

And I like the cut of your jib!(Ah, you're from Florida. No wonder you recognize a Mac. It's on their migratory route.)

 

[Atfirst]

Interesting question. The answer is a plain NO however, no way you are ever going to reach 15 knots ! I own a 31 foot sailboat, that weighs 3.2 tons and has an inboard diesel of 18 hp. In flat water with no sails up I make about 5.5 knots with the engine running at about 2000 rpm. Under sail only, sailing close hauled in a 15 knot wind I make about the same speed. When motor-sailing in the same condition, that is sailing at 5.5 knots and having the engine running at 2000 rpm I will certainly go faster but hardly more than 1 knot. So the propeller most definitely does not cause a drag under these conditions. During maintenance of the engine (and being securely tied to the pontoon) I sometimes run the enigine at full throttle. I have never measured the speed of the water that is being ejected from the back of the boat but it is by all means in excess of 10 knots and probably substantially more. I conclude therefore that if I could sail at e.g. 7 knot and then start the engine at full throttle I would still experience a forward thrust from the propeller but I suspect I wouldn't gain much more than half a knot, if that much.

 

[DaveC426913]

Interesting question. The answer is a plain NO however, no way you are ever going to reach 15 knots ! I own a 31 foot sailboat, that weighs 3.2 tons and has an inboard diesel of 18 hp. In flat water with no sails up I make about 5.5 knots with the engine running at about 2000 rpm. Under sail only, sailing close hauled in a 15 knot wind I make about the same speed. When motor-sailing in the same condition, that is sailing at 5.5 knots and having the engine running at 2000 rpm I will certainly go faster but hardly more than 1 knot. So the propeller most definitely does not cause a drag under these conditions. During maintenance of the engine (and being securely tied to the pontoon) I sometimes run the enigine at full throttle. I have never measured the speed of the water that is being ejected from the back of the boat but it is by all means in excess of 10 knots and probably substantially more. I conclude therefore that if I could sail at e.g. 7 knot and then start the engine at full throttle I would still experience a forward thrust from the propeller but I suspect I wouldn't gain much more than half a knot, if that much.

Couple of things:

1] The numbers in the OP are arbitrary. All of us sailors know that 10 knots under sail is unrealistic. Doesn't affect the question.

2] Your particular sailboat is not designed to go fast under power. Some saiboats are designed to exceed 20 knots under power. Indeed, people can, and do, use them for waterskiing.

 

[Hercuflea]

I hope everyone who responded to this thread is still around!
I have no experience sailing, but let me try to model the physics of the situation.
Suppose the boat is traveling on calm water with a speed $v₀=10$ knots in the positive $x$ direction.
Now drop a small motor into the water and turn it on at full power (the motor can propel the boat in the same conditions at 5 knots under it's own power).
Assumptions:
The wind speed is constant, the boat is traveling at the optimal angle of attack to maximize its speed, and its maximum speed under sail only in these conditions is $v₀$.

The factors that go into determining the boat's speed at this point are:
$F_{s}=$ the force exerted by the sails on the boat at the given wind speed $v_{w}$. (Newtons)
$P_{m}=$ the power output of the motor (kW)
$D=$ the drag force exerted by the water on the hull of the boat, opposite the direction of travel. (N)
$D_{m}=$ the drag force exerted on the motor in the same direction as $D_{h}$ after the motor has been dropped. (N)
$v_{b}(t)=$the velocity of the boat at time t. (m/s)
$v_{a}=v_{b}-v_{w}=$ the apparent wind speed. This is what determines the value of $F_{s}$, not the true wind speed. (m/s)

Now, the motor and the sails have to have a common parameter that we can compare them with. I think the most common way we would describe a sail is the amount of force it exerts on the boat, but the most common way we would describe a motor is the amount of power it outputs. I am not sure how to convert propeller power to force in the forward direction, but I do know from elementary mechanics that

$Power=Force⋅velocity$

So, we can convert the force variables to power variables so that everything has the same dimensions (and are scalars instead of vectors).

$P_{s} = F_{s}⋅v_{b}$,power of the sails (kW)
$P_{m} = P_{m}$ (measurable power of motor, kW)
$P_{d} = (D+D_{m})⋅v_{b}$,power expended by drag forces, kW


The sail-force $F_{s}$ must be some function of the apparent wind velocity $v_{a}=v_{b}-v_{w}$. I'm not sure how to convert incoming wind velocity into force.
The boat's velocity $v_{b}$ is a function of:
The propeller power
The sail force (and thus sail power) (negative sign)
The drag force (and thus drag power) (negative sign)
boat mass $m$
acceleration $a$
Setting the total boat power equal to the sum of the powers:

$$P = P_{s}+P_{m}+P_{d}$$
⇒
$$(∑_{i}F_{i})⋅v_{b} = F_{s}⋅v_{b}+P_{m}+(D+D_{m})⋅v_{b}$$
$$(ma)⋅v_{b} = P_{m}+(F_{s}+D+D_{m})⋅v_{b}$$
$$v_{b}⋅((∂v_{b})/(∂t)) = ((P_{m})/m)+(1/m)(F_{s}+D+D_{m})⋅v_{b}
$$

Is there an easier way to do this? You could just look at the sum of forces itself, but then you need a way to convert motor power in kW to Force in N? Seems like there's an extra unknown somewhere.

(vectors need to have arrow signs but I'm not sure how to add those)

 

[gleem]

I posted an approach in post #29.

The hydrodynamic drag varies about as the square of the speed. so to go from 10 to 15 knots one would need to exert a force of about 2.25 time greater. But the force to move the boat at 10 knots is 4 time that to move it at 5. so it seem you would need 4 x 2.25 =9 times more force or did I just embarrass myself. Now as the boat speeds up the drive from the sails can increase (apparent velocity increases) but this increase in drive cannot be free although I cannot see the connection between the increased sail power and the energy used by the motor.

This can be further extend to the power need.

Speed 5kts 10knt
Net
Force(rel units) 1 4

HP (rel units) 5 40

So how much more speed does 25 hp units buy you if you are already going 10 kts.?

I propose adding 25 to the 160 and determine the speed assuming there is no change in the force speed relationship.

You get 10.4 kts.

YUCK! I would hope that real sailors would not consider this a true sailboat...I think it has a 60 Hp motor hanging off its stern. It is probably a terrible sailing boat. The weight of the engine alone would probably mess up its performance.

Note added: The propeller must be changed from that used at 5 kts since its pitch will be inappropriate for significantly higher speed. You must Increase the pitch.

 

[rcgldr]

To reiterate what I posted earlier, the flow relative to the propeller is different than the boat's speed relative to the water because the propeller operates in the wake of the boat. From Wiki: "Wake is the interaction between the ship and the water with its own velocity relative to the ship. The wake has three parts: the velocity of the water around the hull; the boundary layer between the water dragged by the hull and the surrounding flow; and the waves created by the movement of the ship. The first two parts will reduce the velocity of water into the propeller, the third will either increase or decrease the velocity depending on whether the waves create a crest or trough at the propeller."

http://en.wikipedia.org/wiki/Propeller#Actual_performance

The unknown here is the difference between boat speed relative to water and wake speed relative to propeller.

 

[gleem]

To reiterate what I posted earlier, the flow relative to the propeller is different than the boat's speed relative to the water because the propeller operates in the wake of the boat. From Wiki: "Wake is the interaction between the ship and the water with its own velocity relative to the ship. The wake has three parts: the velocity of the water around the hull; the boundary layer between the water dragged by the hull and the surrounding flow; and the waves created by the movement of the ship. The first two parts will reduce the velocity of water into the propeller, the third will either increase or decrease the velocity depending on whether the waves create a crest or trough at the propeller."

This might be true for an outboard but many sailboats have inboard power with the prop mounted well below the water surface in the non turbulent laminar flow of water aft of the keel and typically two to 10 feet behind the transom and the wake.depending on the size of the boat. So your problem need not be generally addressed.

 

[DaveC426913]

YUCK! I would hope that real sailors would not consider this a true sailboat...

Ah. The ol' No True Scotsman fallacy.

Yup. Well aware of the jealosy that motorsailers engender.

A sailor is someone who sails a boat (unless you're in the Navy or a Church St. bar on Toronto). And there are as many types of sailor as there are sailboats.
Sailboats come in a vast array of configurations to meet every sailor's preferences.
By definition, those who buy a given sailboat, do so because its merits beat those of every other sailboat.

 

[DaveC426913]

... many sailboats have inboard power with the prop mounted well below the water surface in the non turbulent laminar flow of water aft of the keel and typically two to 10 feet behind the transom and the wake...

Sorry, did I read that right?

What I read is you stating that typically, inboards have the prop 2 to 10 feet behind the transom (never mind the wake).

C̶a̶n̶ ̶y̶o̶u̶ ̶s̶h̶o̶w̶ ̶m̶e̶ ̶a̶ ̶s̶i̶n̶g̶l̶e̶ ̶e̶x̶a̶m̶p̶l̶e̶ ̶o̶f̶ ̶a̶ ̶m̶o̶d̶e̶r̶n̶ ̶s̶a̶i̶l̶b̶o̶a̶t̶ ̶w̶i̶t̶h̶ ̶i̶n̶b̶o̶a̶r̶d̶ ̶t̶h̶a̶t̶ ̶h̶a̶s̶ ̶i̶t̶s̶ ̶p̶r̶o̶p̶ ̶a̶f̶t̶ ̶o̶f̶ ̶t̶h̶e̶ ̶t̶r̶a̶n̶s̶o̶m̶ ̶b̶y̶ ̶a̶n̶y̶ ̶a̶m̶o̶u̶n̶t̶,̶ ̶l̶e̶t̶ ̶a̶l̶o̶n̶e̶ ̶2̶ ̶-̶1̶0̶ ̶f̶e̶e̶t̶?̶

No ... you're saying the laminar flow is 2-10 feet behind the keel.
But how does that address whether the prop is in an area of laminar flow?

 

[gleem]

OOps but really you know what I meant, behind looking forward?? Should have said forward of the stern. By the way I am sorry if that was your boat. But that isn't even a proper motor sailer. IMHO.

 

[DaveC426913]

By the way I am sorry if that was your boat. But that isn't even a proper motor sailer. IMHO.

It is my boat. I frequently bask in the glow of jealous sailors, who seem to think their idea of sailing is the only one there is.

Tell you what? Let's go out together one day, and have some fun:

1. We'll sail in ridiculously high winds that knock us over to 45 degrees even when reefed. I'll have helm because my twin keels will still be in the water. What's your max heel before you lose helm?
2. We'll wait until a squall is right on top of us before deciding it's time to head for shelter. I'm there in five minutes. You?
3. Then, when we get bored of this lake by lunch, we'll haul our boats out, and trailer them (behind our 6-cylinder sedan) to another lake for the afternoon.
4. We'll head up a nice cozy river, under a few bridges. I need 11 feet of overhead clearance. You?
5. We'll go right up to the head of the river. My draft is 12 inches. Yours?
6. At sunset, we'll pull right up and beach on the shore of a sandy lagoon and play in the surf.
7. When the wind dies to zero, I'll tow you back to harbour in a half hour.

 

[gleem]

I am glad you are happy with your choice and maybe some astute business man found a niche or larger. but I still say a pox on it. Beauty is in the eye of the beholder and I see none.. I do not see a lot of competition in that market for that concept. I might add that this idea is not new. Back in the early 80's there was an identical idea although I forget the manufacturer, out of business.

So when are you going to the Caribbean ?

 

[gleem]

@DaveC426913 I want to apologize for my elitist/snobbish comments. I have not sailed the boat so I should not have made those comments. It obviously suits your needs and I appreciate its flexibility and advantages. Now back to the OP.

 

[DaveC426913]

No worries at all!

You wouldn't be a sailor if you didn't have strong feelings about boats. It's a requirement!

 

[Hercuflea]

So has this thread come to any consensus at all about the op?

 

[Atfirst]

Couple of things:

1] The numbers in the OP are arbitrary. All of us sailors know that 10 knots under sail is unrealistic. Doesn't affect the question.

2] Your particular sailboat is not designed to go fast under power. Some saiboats are designed to exceed 20 knots under power. Indeed, people can, and do, use them for waterskiing.

Not to the point. Going under sail at 10 knots and adding motorpower that without sails up gives 5 knots will not result in a boatspeed of 15 knots.
Also going faster than 10 knots under sail is a piece of cake for racing sailboats.

 

[A.T.]

So has this thread come to any consensus at all about the op?

Power requirement for a speed increase increases with speed, so you cannot expect the same speed increase at different speeds from the same power.

 

[gleem]

Not to the point. Going under sail at 10 knots and adding motorpower that without sails up gives 5 knots will not result in a boatspeed of 15 knots.

From post 40, I estimated that it would increase the speed 0.4 kts with the provision that the boat is designed to go 15 kts and that the propeller pitch be increased from that used at five knots.to accommodate the much higher speed. To get to 15 kts from 10 kts would require an engine 19 time more powerful than that use at 5 kts. using my approach.

Also going faster than 10 knots under sail is a piece of cake for racing sailboats.

The term racing boats is too generic to make that statement since they vary greatly in design and size too much. 10 knots speed upwind is only obtainable by long waterline racing boats where planing hulls loose their advantage wrt speed but can compensate somewhat by canting keels or shifting water ballast.. Down wind 10 knots readily by smaller planing hull boat and by many displacement hulls under the right conditions.

For your viewing enjoyment I have posted a video of the Volvo Ocean Racing "sled" "Pirates of the Caribbean - Black Pearl" A 70 ft light displacement 31,000 lb planing monohull sailing in a fresh breeze on a beam reach (wind at right angles to the course sailed). my favorite sailing video. The boat has a canting keel that can tilt to windward to help keep the boat upright an important consideration in attaining max speed.This is a boat that can attain 10 knot easily and over 20 down wind. Sorry about any pop-ups.

 

[Cutter Ketch]

This thread seems to have gone on some irrelevant tangents. It makes no difference what the sails do when the motor either speeds up or slows the boat down. It doesn't make any difference what the hull speed of the boat is (except in that clearly it is higher than 10 knots because somehow the boat got there). The original question said nothing about increasing the speed to 15 knots. It doesn't matter how the boat got to ten knots. Etc etc.

The question asked was: given a motor and prop that can propel a boat to 5 knots by itself, if the boat is made to travel at 10 knots will the motor add or subtract? I.e will said motor be a thrust or a drag when it is moving through the water at ten knots. All we are interested in is the sign and all that matters is the motor and prop except perhaps to argue that the 5 knot limit was somehow indicative of something about the design of the motor.

At five knots the thrust from the motor equaled the drag from the boat. That is clearly not zero thrust. The drag from the boat is not zero, so the thrust from the motor is not zero. There is no reason to think the thrust will suddenly drop to zero if the speed through the water is increased. The drag of the boat would increase and the motor doesn't have enough thrust to overcome the increase in drag, thus the top speed, but the thrust cannot and will not instantly drop to zero. if the boat is forced through the water at say 5.1 knots by some other agency whether it be sails, a separate stronger motor, or the boat is being towed behind a battleship, there is absolutely no question that the motor will continue to provide thrust at some speed which is at least slightly higher than the speed the motor alone could achieve.

There is certainly a speed higher than 5 knots at which the answer to the question which was asked is positively yes, the motor will provide extra thrust.

There is a speed through the water where the rotational speed of the prop and the pitch of the prop give a negative angle of attack. The prop cannot provide thrust at a negative angle of attack. There is certainly a speed high enough that the answer to the question which was asked is positively no, the motor will not provide extra thrust. It will be a drag.

Therefore there is a speed higher than 5 knots at which the answer changes from yes to no. The only question then is will that speed be above or below 10 knots.

The only way the boat enters into this question is what it's drag is at five knots. That gives us a single point on the motor's thrust to speed curve, although I'm not sure even that is much help. Everything else about the boat is irrelevant.

Instead of talking about sailboats and hull speeds and points of sail, this question should focus on propeller and motor design.

I feel very strongly that it is possible to create a plausible situation where the thrust of the propeller equals the drag of the boat at 5 knots and yet the thrust of the propeller is greater than zero at 10 knots. So I don't believe that there is a fundamental reason that the propeller can't have positive thrust at ten knots.

I am equally sure that it is possible to create a plausible situation where the propeller turns to drag below 10 knots.

The argument then is not about what is required by first principles, but rather what is likely by engineering choices of the designer of the propeller and motor, not to mention the choices of the boat owner who doesn't always buy the motor and propeller best optimized for how he intends to use it.

There is a perfect example curve here: http://atljsoft.com/html_help/Example, 34' Sailboat.htm

For the weakest prop the thrust crosses the drag a 7 or 8 knots. The thrust curve is rather shallow. They didn't draw much of it and it surely has shape making extrapolation iffy at best, but you can see it is heading off to hit zero at MUCH higher than the 8 knot top speed. Maybe 14 or 15 knots? Could it be double the thrust equals drag point? Sure. Could it be less? Sure.

So despite all the noise the answer is "physics doesn't tell you from the information provided". It can easily be engineered to happen either way, and the thrust vs speed curves I've been looking at suggest 10 knots would be on the edge of where the answer changes from yes to no.

 

[gleem]

The question asked was: given a motor and prop that can propel a boat to 5 knots by itself, if the boat is made to travel at 10 knots will the motor add or subtract? I.e will said motor be a thrust or a drag when it is moving through the water at ten knots. All we are interested in is the sign and all that matters is the motor and prop except perhaps to argue that the 5 knot limit was somehow indicative of something about the design of the motor.

The OP said the motor should be capable of moving the boat at 5 kts.. nothing is said of the prop. We do not have to know anything of the motor just its HP.. If the pitch of the prop is appropriately modified for the higher boat speed and its rpms also appropriately adjusted the engine should incrementally increase the speed. If not but the gear box is changed to increase the prop rpm to an appropriate rate you still might get some thrust albeit less but this I have no idea how to even estimate it. If the engine and prop are left as is then I would expect some drag since the prop would be spinning inappropriately slow for the boat speed.

 

[DaveC426913]

It doesn't matter how the boat got to ten knots. Etc etc.

It matters. Once you turn on the engine and change speeds, it will affect whether the boat can still sail at 10 knots. There is an interplay between the two. It factors into the OP's question, to-wit:

the sailboat is capable of sailing at 10 knots under the current wind conditions using only its sails

Apparent wind conditions change under power.

Instead of talking about sailboats and hull speeds and points of sail, this question should focus on propeller and motor design.

You seem to be ignoring that the motor cannot push the boat to 15 knots. Thus, the fact that it is moving at 10 knots already - factors into the ability of the motor.

 

[Cutter Ketch]

It matters. Once you turn on the engine and change speeds, it will affect whether the boat can still sail at 10 knots. There is an interplay between the two. It factors into the OP's question, to-wit:

No, nothing can change the sign of the contribution. For example if you say the boat speeds up and therefore the sails become inefficient so it slows back down it can't go back through 10 knots (or even to 10 knots) because that is the speed where the sails are more efficient. That's circular logic. More can only mean more. The details of the boat can only affect HOW MUCH more.

 

[Hercuflea]

Let me try to make the assumptions of the OP clear.

1) The boat is sailing on calm water.
2) The boat is traveling downwind at full sail.
3) The wind speed is constant, and given the sail configuration, the current wind speed can push the boat at 10 knots.
4) The boat has a propeller which can propel the boat at a max cruising speed of 5 knots under it's own power.
5) Let's assume the propeller is not lifted in and out of the water when it is on or off. (Although that might be interesting to look at)

OP question: If the motor is turned on full power when the boat is already sailing at 10 knots, will the boat reach 15 knots? If not, what will happen?

Also, I do think it is important that this is a sail boat, because unlike other methods of propulsion, there is an interaction between the motor's power output and the thrust of the sail. The sail's thrust is a function of how fast the boat is going, whereas that wouldn't be the case if it was being tugged for example.

 

[DaveC426913]

No, nothing can change the sign of the contribution.

No one has suggested otherwise.

That's circular logic.

Indeed. The very art of sailing is that interplay between competing factors.

(Example:
more wind => more heel, more heel => more spillage, more spillage => slowing down, slowing down => less heel

A newbie sailor will cycle between fast and slow. A seasoned sailor is one who can balance on the knife edge of that hump.)

More can only mean more. The details of the boat can only affect HOW MUCH more.

Yes. The question was whether motor and sail are 100% additive.

To reach 15 knots, the sail would have to continue to supply 10 knots while the motor provided a cumulative 5 knots, adding to 15.

I grant that boat can exceed 10 knots, it;s just not apparent that it can reach 15.

 

[DaveC426913]

OP question: If the motor is turned on full power when the boat is already sailing at 10 knots, will the boat reach 15 knots? If not, what will happen?

No, it will not reach 15.

The apparent wind speed will drop as the boat is accelerated by motor. The sails wil no longer provide thrust to push it alone at 10 knots.

They are additive, but not 100% additive.

You'll reach some speed between 10 and 15.

 

[Hercuflea]

You'll reach some speed between 10 and 15.

But what about the drag caused by the propeller? If the motor cannot push the boat past 5 on it's own, that means thrust = drag. When the boat is traveling at 10 knots, drag is > than when it is traveling at 5. But theoretically the motor is still outputting the same thrust.

 

[DaveC426913]

But what about the drag caused by the propeller? If the motor cannot push the boat past 5 on it's own, that means thrust = drag. When the boat is traveling at 10 knots, drag is > than when it is traveling at 5. But theoretically the motor is still outputting the same thrust.

Yes. This is what I argued in posts 22 and 25.

In essence, how can a motor (that can only provide 5 knots in calm water) still manage to provide that same 5 knots when in 10 knots of flow?

 

[Hercuflea]

Yes. This is what I argued in posts 22 and 25.

In essence, how can a motor (that can only provide 5 knots in calm water) still manage to provide that same 5 knots when in 10 knots of flow?

I'm not sure if it would provide any knots. It could provide negative knots.

 

[DaveC426913]

I'm not sure if it would provide any knots. It could provide negative knots.

Yes. I can see that rationale, and argued for it.

But I acknowledge that others are raising pertinent factors, and the issue is more complex.

 

[Cutter Ketch]

I must apologize. The OP did originally ask if the boat will reach 15 knots. I did not take it to mean he seriously wanted to know if it will reach 15 knots. It was in the spirit of "what will happen? Will it go 15 knots, will it slow down?" but then I remembered it as "will it speed up, will it slow down".

As I said physics allows for slowing down or speeding up, but yes, you would need to know the details (lots of details) to know by exactly how much.

I agree with arguments that 15 is not possible. (Thrust at 5 equals drag at 5. Drag is nonlinear. Increase in drag from 10-15 is greater than increase from 0-5. Thrust is less at 10 than at 5. 15 is not possible)

Again, my apologies for misremembering the question.

 

[anorlunda]

Man is this thread lost in the woods of complications. Let's try taking @A.T. s advice.

How about looking at the actual physics? Any amount of mechanical power an engine can deliver can be turned into forward thrust at any speed[...]

Think of conservation of energy. Any power that the engine puts out in excess of its internal friction must go to increased boat speed. It can do that by decreasing drag, or by increasing thrust, or both. The only other possibility is that the extra power just churns water increasing the water temperature but not changing momentum of water or vessel.

If we all agree that more power necessarily makes some increment of more speed, that would be an accomplishment. If you don't agree, please explain how you think the energy is conserved.

Next step, how much more speed? That is the place where many complications come into play, and an accurate answer requires a much more detailed problem statement.

 

[Hercuflea]

Man is this thread lost in the woods of complications. Let's try taking @A.T. s advice.
Think of conservation of energy. Any power that the engine puts out in excess of its internal friction must go to increased boat speed. It can do that by decreasing drag, or by increasing thrust, or both. The only other possibility is that the extra power just churns water increasing the water temperature but not changing momentum of water or vessel.

If we all agree that more power necessarily makes some increment of more speed, that would be an accomplishment. If you don't agree, please explain how you think the energy is conserved.

Next step, how much more speed? That is the place where many complications come into play, and an accurate answer requires a much more detailed problem statement.

I do agree with you now. If the motor is fixed under water and the sails provide 10 knots, the energy spent by the motor must go into either forward movement or heat.

What more do we need to specify before we can get a tentative answer on how much? I am really interested to know the answer to the OP.

 

[Cutter Ketch]

hmmm ... that works in the case where the prop is always in the water and the motor is either on or off. I'll agree that the OP did not specify. However the more usual case for sailboats is that the motor is pulled out of the water when not in use. In that case dropping the motor introduces a large drag and the power from the motor has to overcome that before the force is net forward compared to the no motor case.

 

[A.T.]

What more do we need to specify before we can get a tentative answer on how much?

As a first approximation assume that the hull drag = const * v^2 and the propulsive sail force doesn't decrease when you add the motor thrust. Write down the force balance using P = F * v and the above drag model.

 

[anorlunda]

hmmm ... that works in the case where the prop is always in the water and the motor is either on or off. I'll agree that the OP did not specify. However the more usual case for sailboats is that the motor is pulled out of the water when not in use. In that case dropping the motor introduces a large drag and the power from the motor has to overcome that before the force is net forward compared to the no motor case.

No, the more usual case is that sailboats have inboard engines. The props are always in the water. The OP didn't say inboard out outboard.

As a first approximation assume that the hull drag = const * v^2 and the propulsive sail force doesn't decrease when you add the motor thrust. Write down the force balance using P = F * v and the above drag model.

Sorry no on both counts. Reposting from #3

You didn't say which way the wind was blowing.

• It the wind is from behind the boat, then apparent wind speed is true wind speed minus boat speed. Turning on the motor makes the force on the sail less. Enough motor can make the wind force on the sail zero or negative.
• If the boat is heading into the wind (30 degrees is a close to the wind as it can get) then apparent wind is true wind plus boat speed. Turning on the motor increases apparent wind speed and makes the sail more efficient, allowing higher speeds and angles even closer to the wind.

You are also neglecting hull speed. That is a speed at which resistance of the water grows nonlinearly. It is not a brick wall to increase speed but it is very significant. Hull speed depends on the length of the boat, wind and or motor power don't matter.

If you want really rough approximations, consider hull speed as hard max limit on speed. A bit less rough, consider a region where drag grows $v^4$.

IMO, there is no way to make the "how much" problem easy enough to explain in a forum post.