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 (Wd) 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 (Fbd). And that means the dog pushes back on the board with an equal force in the opposite direction (Fdb). 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.
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!