# Explanation for classic mechanics puzzle

I'm trying to figure out a nice way to describe to a kid the physics behind these experiments:

Assuming ideal conditions, we have a small boat with a sale, close to a lake's shore and a fan fixed on the shore, oriented towards the boat's sale. Once the fan is turned on, the boat will start to move slowly away from the lake's shore.

Now, let's replace the fan with a small kid having a long wooden rod in his hand and let's set the boat back in the initial position. Once the kid touches the boat's sail with the rod, the boat will start moving in the opposite direction, slowly...

In the next experiment, the kid notices that his rod is not long enough to propel the boat further away, so he decides to climb in the boat and to try the same thing he did while he was on the shore: push the sail gently with the wooden rod, but, surprise: the boat did not budge an inch.

The kid thought that maybe the rigid nature of the wooden rod is to blame for the failure of the experiment, so he decided to try again, this time by placing the fan on the boat, oriented towards the sail. Did the boat start to move? Why?

Now, I could try to apply linear momentum theory and consider the air as being formed by small beads, each having an impulse $p_i = m_i v_i$, where the masses and the speeds of the particles are roughly equal, and each $p_i$ gets transferred to the sail, but I'm not sure if this gives any reasonable intuition to an elementary school kid.

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Nice thought experiment. Little observation: "impulse"="momentum". –  Cristi Stoica Sep 7 '12 at 13:34
@CristiStoica I guess that would be the proper translation in English for the Romanian word "impuls". I wasn't sure about this. Thanks! –  Mihai Todor Sep 7 '12 at 14:08

It is a quite poor thought experiment. It involves fluid dynamics which is hard to reason about.

Basically you need to surround the sail boat by a surface and integrate the momentum flow through it to find the force. You need to know air velocity field around the boat, which depends on the overall configuration, in order to do it.

I'll show how ambiguous the results may be depending on the configuration. Consider the following pipe configuration:

    ______
<-____  |
| |
____| |
-> F    |
-> A    |
-> N    |
->____  |
| |
____| |
<-______|


It is made symmetrical to avoid angular momentums. Mass conservations tells:

$$\rho u_i S_i = \rho u_o S_o = Q$$

$\rho$ is the density (constant at our speeds), $S$ is the area, $u$ is the flow velocity. Index $i$ means in (fan), $o$ means out.

The force will be equal to the net momentum flux through the system:

$$F = \rho u_i^2 S_i - \rho u_o^2 S_o = \rho Q u_i \left(1 - \frac{u_o}{u_i}\right) = \rho Q u_i \left(1 - \frac{S_i}{S_o}\right)$$

The force might be either sign or zero, depending one the configuration. For the sail things are worth because of the full 3D. For a closer to the sail situation consider:

         ^^
||
|  |
____|  |
-> F     |
-> A     |
-> N     |
->____   |
|  |
|  |
||
VV


Here the system will move to the left.

You should actually do the experiment, like on this youtube video or on the other. Note the results are different.

You may wonder what is the difference when there is wind instead of the fan on the boat. When there is a fan on the boat there is no external pressure gradient. When there is a wind there is a gradient pressure all over the place, which you have to take into account. You may think of pressure as a molecular flux of momentum.

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I wasn't hoping for such a detailed explanation, but it does offer a lot of insight into the experiment. I haven't had any contact with physics in quite a while, so this is a nice refresher. –  Mihai Todor Sep 8 '12 at 12:38
It seems that what you've shown is that if the sail is used as something other than a sail, you can have propulsion from the boat mounted fan. –  Alfred Centauri Sep 8 '12 at 15:42
@AlfredCentauri sail is sail, it just alters the air flow, you need to know how it does this exactly to find the force. Why would you think that all the momentum of the air will be transmitted to the sail? I've illustrated how one does find the force in a simpler 1D cases, they are not sails, they alter the flow in a much simpler fashion. –  Yrogirg Sep 8 '12 at 16:05
That's the difference between a wooden rod and air, wooden rod won't exert forces on anything other then boat, while air will move ambient bodies (other air) as well. –  Yrogirg Sep 8 '12 at 16:09

When the kid pushes the boat with his rod, the rod also pushes back against the kid. In the case where the kid is on the dock, this push-back force pushes against the dock (which is unattached to the boat) through the kid's feet and thus all of the force applied to the boat propels the boat forward. But, in the case where the kid is on the boat, this push-back force pushes against the boat through the kid's feet and thus we have two forces pushing the boat, one forward and one back. According to Newton's 3rd Law both forces must be equal, thus the boat doesn't move.

In the case where we have a fan on the boat, the fan pushes the air which is unattached from the boat no matter where the fan is placed.

EDIT: As referred to in the comments, if the fan is on the boat we have a similar situation as the kid on the boat and the forces will cancel (assuming all of the forward-force provided by the moving air is captured by the sail - something highly unlikely).

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The fan pushes the air and the air pushes on the blades of the fan blades push the base of the fan and fan pushes on boat. Am I missing something here? –  Argus Sep 7 '12 at 13:39
Actually, you have it right Argus. The question through me off because I read it too fast and assumed that the boat would move forward if a fan, attached to the boat, is blowing air towards the sail. But, now that I think about it - that shouldn't be true... –  mcFreid Sep 7 '12 at 13:42
In the ideal case where the entire "forward-force" provided by the momentum embedded in the air molecules is captured by the sail, how would you explain the fact that this is identical to having the kid on the boat, pushing the sail? He's going to have a really hard time seeing the connection, because the body of air is completely detached from the source, when it hits the sail. Actually, in this ideal setup, the boat might start to slowly move forward, due to the pressure caused by the air molecules that get bounced off the sail. Will it get canceled out by the vacuum forming behind the fan? –  Mihai Todor Sep 7 '12 at 14:27

The crucial point here is internal versus external force.

The fan experiences a force in the direction opposite the flow of air (think of a propeller).

When the fan is on land, this force is not applied to the boat system. The flow of air from the fan produces a force on the sail which accelerates the boat.

If the fan is attached to the boat, the force on the fan is no longer external to the boat system. The fan acts as propeller applying a force to the boat opposite the flow of air. Ideally, it there is a sail capturing the full air flow produced by the fan, the force from the sail cancels the force from the fan. In reality, I imagine that the propulsion from the fan exceeds that from the sail so the boat moves opposite that desired.

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It would be really interesting to see the boat moving backwards, but yes, you're right. On the other hand, if you tell a kid that the air is made of small "balls" that are not "glued" / linked together, he'll have a very hard time understanding what is the actual difference between the two scenarios, since the result is always the same: the balls hit the sail and transmit their momentum to it. I have to explain somehow what happens at the source and why does it make a difference. –  Mihai Todor Sep 7 '12 at 14:19
The momentum of the "balls" cannot have been conjured out of, er, thin air. By conservation of momentum, the momentum imparted to the balls by the fan blades must be offset by an equal and opposite momentum imparted to the fan. This is just an elastic collision problem. Fan blade collides with ball, ball collides with sail. Momentum is conserved. –  Alfred Centauri Sep 7 '12 at 15:32
So, if the fan can magically start at full power instantly, then the boat will move back slightly, until the air reaches the sail. Then it would move forward a bit, until everything balances out. –  Mihai Todor Sep 7 '12 at 16:29
The thing is that there is not just a fan and a sail, there is also an air, that's the full system. And air can take certain momentum so the boat will move. –  Yrogirg Sep 8 '12 at 13:47