# Overview: Where and when?

#### Prerequisites

Our goal in this course is not just to discuss "How do things move?", but to do it quantitatively — mapping the physical phenomenon of motion onto mathematics.

To do this, we have to define ways of associating mathematical structures with both where something is (position) and when it is (time). Since our experience with both location and time matches the arithmetic of the real numbers well, we'll map both physical position and time onto numbers.

Since everyone has had experience with giving numbers to how far away something is or how long something takes to happen, this seems rather trivial. But since we want to inherit the operations of mathematics, we have to be careful to create an unambiguous process -- one anyone can recreate, so we can communicate clearly and precisely with each other.

The process of assigning a number to physical quantities by a measurement is called an operational definition. We specify the operations necessary to create the number (or numbers) assigned to the quantity and what kinds of quantities they are. (See Measurement and Dimensional analysis)

Paying attention to how our numbers are created by measurement really helps us not fall into traps, such as confusing location with length or acceleration with velocity. Here are links to the discussions of the most important principles and foothold ideas.

• Location in space - Coordinates —  Specifying "where" requires building a mathematical structure to locate something: a coordinate system. Since we live in 3D, this requires more than one number.
• Vectors — The mathematical structure we use for specifying everything about motion is the vector. This requires building some tools and methods and a bit more algebra.
• Time — Specifying "when" requires a way to measure time: a clock.
• Kinematic variables — Following an object's position through time gives possibilities for other ways to talk about motion using the derivatives of position with respect to time. These turn out to be crucial in understanding the causes of motion.
• Velocity — How position changes with time: its first derivative
• Acceleration — How velocity changes with time: the second derivative of position
• Kinematic graphs and consistency — As we build multiple ways to describe motion in time, we will have multiple graphs to describe the same physical situation. This is a common practice in science and can be quite challenging. Motion, with which we have direct physical experience, it an excellent place to develop the skill of reading, creating, and interpreting multiple graphs.

Joe Redish 2/10/19

Article 306