An example that may be familiar from school is the problem of determining the motion of a ball thrown by hand. We can use this simple problem to illustrate the increasing complexity of models based on the assumptions and the questions asked.

Our first model relies on Newton’s equations of motion that states: the mass m times the acceleration avert of a point‐mass in the vertical direction is equal to the vertical force fvert applied to the object (often summarized as fvert = mavert). Similarly, we know that the acceleration in the horizontal direction is related to the force applied to the mass in the same direction by fhor = mahor. By assuming that the ball is a point‐mass subject to a constant force of gravity, and neglecting friction, the horizontal force vanishes and the vertical force is constant. Then recalling that the acceleration is the second derivative of the position with respect to time, the model takes the mathematical form of two second‐order linear differential equations for the vertical and horizontal position of the ball as a function of time. The variables in our problem are the positions and the parameters of our problem are the mass, the initial position and velocity, as well as the acceleration of gravity and time. Then, a simple computation will determine the time taken before hitting the ground as well as the distance travelled. The solution of the model would also describe the velocity and the kinetic and potential energy of the ball at all times. In other words, this model can be solved exactly and completely.

The astute reader would note that this ideal description is not very realistic, as the air offers a drag resistance to the motion of the ball that we have neglected.A second model would then take into account the effect of air friction through a simple phenomenological law characterizing the force slowing down the ball. Here, the force acting on the ball is the sum of the force of gravity and a drag force proportional to the velocity. This equation is still simple enough so that it can be solved by traditional methods. This new model must contain an extra parameter, the drag coefficient, and its solution provides better estimates for all kinematic and mechanical quantities.

A perceptive reader might then object and correctly question the assumption that the force acting on the ball remains constant. Indeed, as the ball’s vertical position increases, the force acting on it decreases according to Isaac Newton’s law of universal gravity, which states that the force acting between two masses varies as the inverse of the square of the distance between their centre of gravity. While more complicated, a third model taking into account such a minute effect can easily be written down and solved. The analysis of this model will reveal that this effect is indeed negligible for the motion of a ball but that it may be relevant if one tries to understand the motion of satellites or inter‐continental ballistic missiles.

Yet, the sporty reader is aware of another effect. When a cricket or baseball ball is spun during the throw, its position can wobble in the air, an effect that none of the previous models describe. In this case, experience (rather than experiments) guides us in considering a fourth model that will take into account the motion of an object of finite size that includes its spin and the interaction of the air with the stitches of the ball. This problem can be formulated but its solution is extremely complicated. A combination of advanced mathematical methods and extensive computational studies of such problems, coupling the air flow with the motion of the ball, has revealed how such balls can change direction multiple times in the air when properly spun. These methods, combined with experiments in wind tunnels, are used to design new sports balls.

Finally, a refined reader could further object that as the relative position of two masses change, they affect the underlying structure of space‐time in which they sit by changing the metric used to measure both the distance between objects and the time interval between events. The problem of throwing a ball should then be solved within the theory of general relativity. A fifth model taking these effects into account could, in principle, be derived. But before doing so, it is important to question the relevance of such an approach. Modelling is not an exercise in abstract physics. It must rely on sound judgement and on the understanding of the underlying problem. In our case, unless the Earth and the ball have masses comparable to that of the Sun, we know from simple estimates that these effects do not play a role, and, indeed, could never be measured.