148 — Do Dogs Know Related Rates Rather than Optimization?

If you throw a ball into a lake and you have a dog that is into this sort of thing, it is likely that you will watch your dog bound down the shore and leap into the water im order to swim to the ball.

What is interesting is that — with great likelihood — your pupper will not run directly at the ball, describing the shortest path; nor will your pupper run parallel to the lake shore until even with the ball, and then turn 90º to swim (minimizing the amount of swimming required)… Instead, such a floof would likely run along the shore to some point before the ball, and then turn and swim at an angle to reach the ball, thus describing the most time-efficient path to reach the ball.

This function is described as:

where $r$ is the running speed of your dog, $s$ is the swimming speed, $x$ is the distance of the ball from the shore, and $y$ is the distance from the ball, measuring parallel to the coastline, at which your dog veers into the water.

It seems unlikely that dogs are computing calculus in their heads while running; instead, Perruchet & Gallego and the Labrador Salsa (reproducing earlier work by Pennings and dog Elvis) demonstrated that for any dog that knows its running and swimming speeds, an instantaneous, moment-by-moment calculation will result in the same optimal-path formula. That is, instead of computing a global solution, it is more likely that Elvis and Salsa are computing many small local solutions, which closely approximate the optimal path — as is the wont of calculus.