Water skiing is a sport where an individual is pulled behind a boat or a cable ski installation on a body of water, skimming the surface.
Consider an idealized case where the boat is moving at a constant velocity $\overrightarrow v_0=\text{const}$ (relative to the water), independently of the skier. The boat and skier are connected by a massless, unstretchable rope. Surface of the water is assumed to be smooth.
Then, what is the maximum possible (instantaneous)speed of the skier $ v_\text{max}$ (relative to the water)?
My first guess, based on intuition, is that $v_\text{max}=2v_0$.
But I'm not at all sure.
Edit:
Skier's velocity may vary. Consider vector projections of velocity vectors $\overrightarrow v_1 $ and $\overrightarrow v_2 $ of the boat and skier in the direction of the rope. Because the rope is unstretchable, these projections must be equal. That means, must hold the equality: $$v_1\cos{\alpha}=v_2\cos{\beta}$$
$\alpha$ is an angle between $\overrightarrow v_1 $ and the direction of the rope
$\beta$ is an angle between $\overrightarrow v_2 $ and the direction of the rope
So for example, if $\alpha<\beta$ then $v_2>v_1$. I.e. the skier is moving faster than the boat relative to the water surface.