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In the theory of heat of the first half of the nineteenth century, heat was a substance. Mayer and Joule contradicted this thesis but developed different concepts of heat. Heat was a force for Mayer and a motion for Joule. Both Mayer and Joule determined the mechanical equivalent of heat. This result was, however, justified in accordance with those concepts of heat. Mayer's characterisation of force reappears in the very common textbook definition "energy cannot be created or destroyed but only transformed" and his theory led to a phenomenological approach to energy. Joule and Thomson's concept of heat led to a mechanistic approach to energy and to the common definition "energy is the capacity of doing work". One and the same term "energy" subsumed these two approaches. The problematic concept of energy, energy as a substance, appears then as a result of an eclectic development of the concept. Another approach, which appeared in the 1860s, is directly based on the mechanical equivalent of heat and can be characterized by the use of "principle of equivalence" instead of "principle of energy conservation". Unlike the others, this approach, which has been lost, poses no problems with the concept of energy. The problems with the energy concept as to the kind of phenomena dealt with in the present paper can, however, be overcome, as we shall see, in distinguishing between that which comes from experiments and that which is an interpretation of the experimental results within a conceptual framework.

It is generally accepted nowadays that History and Philosophy of Science (HPS) is useful in understanding scientific concepts, theories and even some experiments. Problem-solving strategies are a significant topic, since students' careers depend on their skill to solve problems. These are the reasons for addressing the question of whether problem solving could be improved by means of HPS. Three typical problems in introductory courses of mechanics--the inclined plane, the simple pendulum and the Atwood machine--are taken as the object of the present study. The solving strategies of these problems in the eighteenth and nineteenth century constitute the historical component of the study. Its philosophical component stems from the foundations of mechanics research literature. The use of HPS leads us to see those problems in a different way. These different ways can be tested, for which experiments are proposed. The traditional solving strategies for the incline and pendulum problems are adequate for some situations but not in general. The recourse to apparent weights in the Atwood machine problem leads us to a new insight and a solving strategy for composed Atwood machines. Educational implications also concern the development of logical thinking by means of the variety of lines of thought provided by HPS.

There has been much research on principles and fundamental concepts of mechanics. Problems concerning the law of inertia, the concepts of force, fictitious force, weight, mass and the distinction between inertial and gravitational mass are addressed in the first part of the present paper. It is argued in the second that the law of inertia is the source of these problems. Consequences drawn from the law explain the metaphysical concept of force, the problematic concept of fictitious force, the nominal definition of weight and the difficulty with defining mass operationally. The core of this connection between the law and these consequences lies in the fact that acceleration is a sufficient condition for force. The experimental basis of the law in the course of its history shows, however, that the law presupposes acceleration necessarily whereas acceleration does not presuppose the law. Therefore, there is no inconvenience in taking acceleration independently of the law. This is enough to bypass those problems. Taking into account how force is measured by force meters and how mass is basically determined, by comparison with the standard mass, a minimal meaning for both concepts of force and mass is established. All this converges with several solutions proposed in the course of history and increases the communicability of mechanics, as outlined in the final part of this paper.

Some physicists have pointed out that we do not know what force is. The most common definition of force in textbooks has been criticized for more than two centuries. Many studies have shown that the concept of force is a problem for teaching. How to conceive force on the basis of the concepts and criticism of force in the works of Newton, Euler, d'Alembert, Lagrange, Lazare Carnot, Saint-Venant, Reech, Kirchhoff, Mach, Hertz and Poincare is the question of the present article. This part of the article is followed by an overview of definitions of force in contemporary textbooks. In the next part, an answer to the question is given: how to understand force within the framework of the laws of motion and in applications. Finally, some educational implications are considered.

Some physicists have pointed out that we do not know what energy is. Many studies have shown that the concept of energy is a problem for teaching. A study of the history of the concept shows that the discoverers of energy did not find anything which is indestructible and transformable but rather that the concept of energy underwent a change of meaning and energy was considered a substance towards the end of the nineteenth century. In distinguishing between the treatment of phenomena and the theories carried out by Mayer and Joule, it can be concluded that they established equivalences between different domains, such as motion and heat, motion and electricity or position and motion. This complies with the interpretation presented in textbooks published about a century ago and enables us to overcome some difficulties with the concept of energy.

The law of inertia is a problem in teaching due to the impossibility of showing the proposition experimentally. As we cannot do an experiment to verify the law, we cannot know if it is correct. On the other hand, we know that the science based upon it is successful. A study in the history of mechanics has shown that there are different foundations for the law but also that the law plays the same role in the science since Newton. To avoid a statement of which we cannot be sure, the present paper proposes to understand the law through its function in the theory. In this case, we do not have to say how a free body moves, but rather that the rectilinear and uniform motion is the motion of reference in Newtonian mechanics.