UPDATE: Here is my answer. Enjoy! I'll post another question like this in a day or two.
We've been providing good math and physics problems that we hope everyone enjoys. Above and beyond these problems, which generally require application of known techniques, there is a set of deeper questions about physics and why some math appears but not others. This is the domain of theoretical and mathematical physics. Understanding "why this math?" can lead to a much deeper understanding of the fundamental physical principles of our world.
To further this goal, I'll be asking questions on seemingly simple equations in physics that have deep physical reasons why they are those equations and not others. I hope that our bright users spend some time thinking about them and discover the answer, and, if they prove popular, I'll chime back in after a day or two with a link or post about the answer and pose another question.
Our first question:
Newton's laws (in 1-d) state \(F=ma\), where a is the second time derivative of the position. Why does this equation have two time derivatives and not, say, 1,3,4, etc. time derivatives? In particular, what are the physical reasons behind this?
Note: Answering why this rule exists with another mathematical rule, like "because we work off Lagrangian mechanics and those have single time derivatives" is not okay - it just replaces one rule with another. We want the physical reason...