Introducing the harmonic mean solving a tourist’s problem

This exercise can be given to a group of students with basic knowledge of mathematics and physics at the beginning of a lesson. We can imagine that there will be some students that will solve the exercise using the “common sense” solution recalling basic notions of physics and some students that will solve the exercise recalling basic notion of physics and computing a mean velocity using the arithmetic mean, the most common mean. At the end of the exercise, the teacher will compare the two solutions and will present the harmonic mean as the fastest solution for the students that solved the problem using the “common sense” solution and as the correct mean to be used for the students that solved the exercise computing a mean velocity.