## Mathematical Modeling in Physics

One of the features that defines physics as a discipline is the practice of writing ideas in mathematical forms. Physicists favor mathematical models above all else, for their precision and ability to make quantitative, testable predictions. Ideally, through the process of teaching students to build, test, and revise mathematical models, they will gain some deeper understanding of the discipline of physics and how physicist use mathematical models to understand the world.

To a physicist, an equation is a representation of a scientific model, an explanation describing how or why a system behaves the way it does. These mathematical models can play various roles in physics:

• Identifying causal links. In many physical systems, there is an interaction between two objects that causes a change in the system. An expert can identify causal links in a system by looking at a mathematical model. For example: A net force acting on an object causes an acceleration (Newton’s 2nd Law), or a current moving through a wire causes a magnetic field to be generated (Ampere’s Law).

• Making sense of complex systems. Another way we use models is to make complicated problems easier to understand. To do this, scientists have invented abstract mathematical constructs that help us predict and track changes in a system. For example, energy is not a real thing that I can hold in my hand; it is an abstract concept that was developed to understand and describe changes in a system. In the same way, electric and magnetic fields are abstract concepts that help us to conceptualize how charged particles behave.

• Describing the behavior of a system over time. Fundamental relationships, such as Newton’s Laws, can be used to predict the behavior of a system over time. The “equation of motion” predicts where the object will be at a given point in time in the future, or where it was at some point in the past. This one equation tells lets us predict everything about the motion of an object, given the initial conditions of the system. At the introductory level, the kinematics equations are a good example of how we use mathematical models to make predictions (e.g. With what speed will the ball hit the ground? Where will it land?).

To get students to engage in the practice of treating equations as mathematical models that are representations of scientific ideas requires constant reframing of how we talk about equations in the classroom. One way to engage students in this mindset is to have them constantly building, testing, and revising mathematical models.

For example, we reframed the first unit in our introductory physics courses as “Modeling Motion.” We include all of the standard parts of the curriculum (graphing, vectors, kinematics equations), but we now present it as a model building exercise. The students make a series of videos throughout the unit, and use video analysis software to track the motion of various objects. They develop increasingly sophisticated empirical models for motion based on the graphs, and the kinematics equations are introduced at the end, after they have “discovered” free fall on their own. The final project for this unit is for the students to perform a video analysis of a two-dimensional motion of their choice (e.g. a dropped coffee filter, a remote control car). The final report requires them to build an argument using multiple representations (vectors, diagrams, graphs, mathematical models) as to whether or not the motion undergoes constant acceleration.

On a smaller scale, an individual lab can be turned into a modeling exercise. For example, a standard projectile motion lab asks students to fire a projectile and hit a target. This can be reframed in terms of modeling by asking the students to build a mathematical model that can predict where the ball will land. This requires students to find the initial conditions of the projectile, and then solve for the position where the ball will land. Students can also make a computational model in Excel, where they can easily change the initial conditions and see how their prediction changes. Finally, they conduct the experiment and check to see if all three models (theoretical, computational, and empirical) agree.

On an even smaller scale, standard problems can be rewritten as modeling exercises. For example, take this typical back of the book problem:

A coin rolls along the top of a 1.33 m high desk with a constant velocity. It reaches the edge of the desk and hits the ground 1.00 m from the edge of the desk. (a) What was the velocity of the coin as it rolled across the desk? (b) What is the final velocity of the coin (magnitude and direction)?

We can take the same prompt, and step the students through a modeling approach to solving the problem:

A coin rolls along the top of a 1.33 m high desk with a constant velocity. It reaches the edge of the desk and hits the ground 1.00 m from the edge of the desk. (a) What assumptions can we make in modeling this motion? (b)What are the known initial conditions of the coin? (c) What questions can we answer about the motion of the coin? (d) Find the answer to one of your questions.

All of these curricular changes do take time and effort. In reality, the simplest thing that instructors can do is to emphasize modeling as they present new material. Just by introducing the concept of mathematical models as a form of scientific models, or using the word “model” instead of “formula” or “equation”, can make a difference in how students view the process of solving problems.