What to Expect From a Pooch on a Paddleboard

It's great when you can combine two things you enjoy—like paddleboarding and spending time with your best furry friend. But what if you add a third great thing into the mix? What if you add some physics? In that case, someone is going to get soaked.

That's what happens in this funny video . Check it out. A woman is out on a paddleboard with her dog. As they approach the shore, the dog sees something that demands immediate canine attention and leaps for dry land. But in doing so, he causes the board to shoot back, and ... splash . Why does that happen? Let's figure out the physics!

How Things Interact

There are two key ideas we need here. One is the momentum principle. First, remember that momentum (p) is defined as the product of an object’s mass (m) and its velocity (v). The faster something goes, or the bigger it is, the more momentum it has.

Illustration: Rhett Allain

So, now, the momentum principle says that a net force on an object changes the object’s momentum. As an equation, it looks like this:

Illustration: Rhett Allain

The other key idea is that forces are an interaction between two objects. There must be two objects for there to be a force. If you push on a wall, the wall pushes back on you with a force of equal magnitude. It's not that the wall resents your mistreatment, that’s just how it works. These two forces are really just two ends of the same action.

Conservation of Momentum

OK, now we’re ready to analyze this pooch-paddleboard incident. For now, let’s represent this system as if it consisted of just two objects—the dog on one hand and the paddleboard-plus-human on the other. Although the board is initially moving forward, we'll assume it starts at rest. It doesn’t affect the result.

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Illustration: Rhett Allain

As you can see, each of these objects has three forces acting on it—two vertical forces and one horizontal force. The vertical forces aren't that important here. For the dog, there is its downward weight (W d ) and the upward “normal” force from the board. Same for the board-plus-human. (Technically, the upward force on the board is a buoyancy force from the water—but if you want to let the b represent the board, I'm fine with that.)

But let’s zero in on the horizontal forces. First, there is the frictional force that the board exerts on the dog. Since the dog wants to increase its speed (and thus its momentum), there is a force pushing the dog forward (F bd ). And that means the dog pushes back on the board with an equal force in the opposite direction (F db ). Right? Two ends of the same interaction.

Of course, since both objects have the same magnitude change in momentum, but the dog’s mass is much smaller, the dog will have a much larger change in velocity.

This situation is an example of conservation of momentum . This says that whatever the momentum of everything is before the dog starts running must be equal to the total momentum after the dog starts running.

This conservation of momentum comes straight from the momentum principle. But it only works if there are no significant external forces on the whole system (consisting of the dog, the board, and the woman). That’s the case here, since the board is in the water.

If you repeated this situation but put the board on dry ground, the frictional force between the board and the ground would prevent the board from changing momentum. Of course, the dog didn't think through the physics here—or maybe he did and just thought it would be a good joke.

Slip n' Slide

You don't really understand something until you can reproduce it with an experiment. So, that's what I did. First, I put two low-fraction carts on a track (in the video below, they’re the flat blue carts in the shadow). Then I put another track on top of them. Now this top track can slide back and forth with relatively little friction—like the paddleboard on the water.

On top of that are two red things: another low-friction cart representing the woman and an electric-powered buggy with a remote switch for the dog. When the buggy accelerates to the left, this causes a force to push on the movable track, making it accelerate to the right. Check it out.

Video: Rhett Allain

Into the Drink

So far, this is all fine with the woman. But that's because we considered her and the board to be one thing, and clearly they are not. From her perspective, the board is accelerating backward, and since only her feet are interacting with it, her feet are yanked back horizontally by the frictional force. This foot-force exerts a torque about her center of mass and makes her rotate forward. It only takes a little bit of rotation until her center of gravity is no longer above her feet—whoopsie daisy!

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Wait! What about Newton's third law? It says:

"For every action there is an equal and opposite reaction."

Yes, the dog moves one way and the board shoots back the other way. But really there is something bigger here—it's the idea that forces come in pairs. Forces are always an interaction between two things. The dog pushes on the board, and the board pushes back on the dog with a force of equal magnitude. Using this with the momentum principle explains the woman's eventual splash landing.

Oh, the “action” and “reaction” definition can be somewhat misleading. The wording implies that there has to be something moving for the forces to be equal—but that’s not true. If you are at rest on the Earth (I'm not judging, just assuming you are on Earth), then the Earth pulls down on you with the gravitational force. But you also pull up on the Earth with an equal magnitude force. You probably didn’t know you were so powerful! It doesn't matter how massive the objects are—there are always two sides to the story.

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