Explaining to a student what we mean by "let the variable $x$ be [some physical quantity]" I recently had a student in a multivariable calculus class I am instructing (as a teaching assistant) ask the following when we started discussing the applications of the course material in physics:
what do mathematicians mean when they say something like "let the variable $t$ be time, or let the variable $x$ be the position of something, etc.?"
My response to this question was something along the lines of "we mean that we're denoting the physical quantity time by the variable $t$." Admittedly, this response felt a little unsatisfying and too short for a question the student had genuinely thought about. With that said, is this an appropriate reasoning (or answer for a curious student) for what we mean when we say something like "let $t$ be time" or, as another example, "let $m$ be the mass of a ball"?
 A: This question can definitely be interpreted differently by different people, so I can only answer from my perspective. Additionally, without talking with the student, it is difficult to address their specific concerns.
In my view, the pure mathematician doesn't mean anything more than "let $t\in \mathbb{R}$" or "$x\in \mathbb{R}^3$." It is the physicist, or applied mathematician, who claims that time exists in some sense and can be modeled by the real numbers, or that space actually is out there and can be modeled by a three dimensional vector space over the real numbers ($\mathbb{R}^3$).
If you can get the student to agree that there is something out there called "time," they will likely agree that (atleast locally) it has a lot of the properties of the real numbers. A mathematician works with mathematical concepts that exist in their minds, while a physicist claims that these concepts are useful models of real world objects. Even physical laws, like Newton's laws of motion, are just mathematical models with specific properties until you make the claim that there are things in the world that behave according to these laws. In most cases, these models will break down at some point, so you need to use more complex models; for instance, you can model a projectile by a point mass, but in many cases, more insight is gained by modeling the object as mass distribution.
A: When thinking about and explaining these applications I find it useful to imagine the underlying set $S$ of all possible physical states of the system. Then "variables" like $x, v, m, t$ are essentially real valued functions with domain $S$ - position, velocity, mass, time. Their values are related by the equations expressing the physics. Then the chain rule is often your friend.
More formally, $S$ is usually a manifold, not just an abstract set, but you don't want to go there in multivariable calculus.
