Best mathematical object to describe speed

Question

Suppose we want to describe speed of a particle, moving between two points on a real line:

---------------$0$----------------$1$-----------------

• If the particle starts at $0$ at $t=0$ and moves left, its speed is negative.

• If particle is constantly in 0, its speed is zero.

• If the particle starts at $0$ at $t=0$ and moves right, its speed is positive.

• If particle is simultaneously at $t=0$ seen in both $0$ and $1$, its speed is infinite.

• If particle starts at $1$ at negative time and moves left, reaching $0$ at $t=0$ its speed $\gt \infty$.

I wonder, whether non-standard numbers, such as hyperreal numbers would be appropriate here?

Answer 1

To respond to the item If particle is simultaneously at t=0 seen in both 0 and 1 , its speed is infinite, what you seem to be describing is a particle whose position as a function of time is given by the Heaviside function. Then in the terminology of the physicists its velocity is given by the Dirac delta function. The hyperreals are indeed useful in providing a mathematical formalisation of both the Heaviside function and the Dirac delta; see for example this recent article.