Now, I don’t know exactly what’s going on, because apparently we’re flying in the face of Newton’s laws
Grant Imahara

So according to the Mythbusters, they violated Newton’s laws in this episode.  Presumably they don’t really believe this, or else they would be on their way to Sweden to accept their Nobel prize.  I have to say, I find it a bit odd that they bring in experts to consult on sound effects, but not physics; rather they let Grant make stuff up.  At least they have enough honesty for him to preface his comments with “I don’t know what I’m talking about”.

All right, here’s a go at an explanation.  Us fluid dynamicists have a great tool for these types of analyses to keep us from getting all wrapped up in the details: control volume analysis.

$\tfrac{\partial}{\partial t} \int \rho \vec{v} dV + \int \rho \vec{v}(\vec{v}\cdot \vec{n})dA=\vec{F}_{ext}$

Basically, by looking at the air into a control volume compared to the air leaving a control volume, we can calculate the force on the air in the control volume, and by Newton’s third law, the force on the objects in the control volume.

So a normal sailboat, going with the wind, will “catch” the air blowing into the control volume in the direction of motion, converting the momentum of the air into its own momentum.  In an unmodified swamp boat (or a jet skateboard), air is leaving the control volume in single direction; therefore, there must be a force on the air in that direction; therefore, there must be a force on whatever is moving the air in the opposite direction.  At the end of the day, in both cases air is being accelerated to the right, propelling the craft to the left.

When fan is turned around and the sail is put up, the air is redirected, mostly to the sides.  if the air leaves the control volume more or less equally in all directions in the plane of the sail, the area integral will sum to zero (contributions in opposite directions will cancel each other out).  If the fan is blowing hard enough (and the sail cooperates), I can imagine that the air would be deflected back towards the fan and leave the control volume to the right, resulting in the observed forward thrust.

Note, this also explains why the swampboat was turning wildly, because the air was favoring one side of the sail or the other, and leaving our little control volume in an unbalanced manner; imparting a sideways force.

I tried to think of a good non-fluids analogy, and here’s the best I could come up with.  Hopefully, it is relatively intuitive that if you were standing on a skateboard or whatever, and threw tennis balls off of it in a particular direction, you would move in the opposite direction.  Newton says so!  Specifically, conservation of momentum.  In order to maintain your initial zero momentum, the momentum you impart to the balls will be canceled out by your momentum in the opposite direction.

Okay, if you’re still with me, imagine erecting a board on the skateboard, and now throw the balls at the board.  If the board was very sticky, and the balls didn’t rebound (or deflected sideways), then they would have the same velocity as the skateboard, which would have to be zero to maintain the system’s initial zero momentum.  If they reflected perfectly elastically, they would have the same velocity that they were thrown at, completely reversing the acceleration of the skateboard to compensate (this would be equivalent to throwing the balls the other way, straight off of the skateboard).  Say they rebounded less than elastically, or sometimes rebounded to one side or the other; then we’d have a situation more analogous to the fan-powered sail; thrust in the “right” direction, but not nearly as powerful as if the sail wasn’t there.

This isn’t a perfect analogy, because the tennis balls are traveling on the skateboard before they get thrown, whereas the air that the fan is blowing is being pulled from the surroundings.  That being said, since the air is being pulled more or less from all directions, it is not contributing significantly to the thrust.

If the fluid dynamics of sailing is something that interests you, check out how to sail into or faster than the wind!

### 6 Responses to “Blow Your Own Sail”

1. Steve says:

Hey,

I understood this using the control volume explanation, but can you explain this using Newton’s laws? It is killing me and I can’t go to sleep.

Would the force on the fan (pushing the boat backward), can the force on the sail (pushing it forward)? And what is the relevance of the big sail as opposed to a normal sail?

2. Steve says:

Actually I think I get it now after looking the tennis balls, but is it because the sail does not absorb all the energy of the wind? If it did absorb all the energy of the wind, then the wind would not bounce back, in your analogy, ‘ a sticky sail’.

• Ryan says:

Steve, I think you’ve got it (sorry I didn’t see your comments sooner; too much SPAM to weed through!)

That’s why I included the tennis balls bit, because I know that control volumes don’t sit well with everyone.

Like you said, if the sail didn’t “absorb” the wind, or just scattered it, the boat wouldn’t move. BUT, if it sort of redirected it backwards, the boat would move forwards. I think this is why they needed a big sail; it had to be big enough to make the aerodynamics work that it would more or less cup the fan’s blown air, and reverse its direction as it left the sides.

Not sure if this ASCII art diagram will help

/      –> redirected back
/    /
|     \
|       < ----- from fan |     /  \     \    \      --> redirected back

^ sail

3. Johan says:

The reflected thrust will contribute, but it can never be larger then the thrust of the fan. What about air spilling around the edges (especially the rigid top and bottom and the accellerating across the forward surface of the sail? Then Mr. Bernoulli’s mechanics would kick in and generate a pressure difference over a larger area (note how the myth proving worked when the sail was made larger?) thus generating a large enough force to overcome the thrust of the fan. My technical knowledge is limited, but I have it that in marine and aero screws (propellers) the “broader bladed ones are more efficient?

4. Anthony says:

After seeing the mythbusters episode I wondered if a pressure gradient had anything to do with it as well. The fan blowing on the sail would have produced a higher air pressure on that side of the sail, and a low pressure on the opposite side. Would this have caused a suction effect to the low pressure causing forward momentum?

• K-Mob says:

That is correct. However, this is largely offset by the force the fan puts on the boat as is sucks air toward it. That’s why they don’t get much of a head of steam going. Of course, as we see, by redirecting the air, you can get some push forward.