Exploring Negative Time in Motion Equations – Need Insights!

[nimble storm]

I’ve been working on a thought experiment related to the roots of negative numbers and motion equations. While analyzing two runners, Alice and Bob, I came across an interesting result that I’d love to discuss.

The Setup
I have two graphs:

The first graph represents how Alice and Bob move in real life, with their respective velocities and accelerations plotted. Their paths are shown, and I derived an equation to determine when they meet.
The second graph is a hypothetical scenario where their paths extend into negative time. This setup suggests they would meet at
𝑡 = − t=−6 seconds, implying a quadratic equation of the form:
𝑡^2 − 6 = 0

The Question
When solving for their meeting time using the first graph (the real-world scenario), I get two solutions:

𝑡 = 0 (which makes sense)
t=−6 (which seems impossible in real life)

However, this negative time solution appears naturally in the second graph. This made me wonder:
Is there a way to derive an equation directly from the first graph that inherently gives both solutions?
Does the equation depend on their starting positions?
Am I possibly working on a problem with an impossible solution, or is there a deeper mathematical reason for the negative time result?
I’d love to hear thoughts on whether the negative time result has physical meaning or if it's just an artifact of solving the equation mathematically. Let me know what you think!

[shrewd barn]

Exploring Negative Time in Motion Equations – Need Insights!

[austere geyser]

animeonfire

[sick temple]

Although it may hold different interpretations in different systems, in deterministic systems (systems like above where newton's laws can be followed for relative motion) a negative time refrences events that have occurred before the point from where the system is believed to have started.
In your example, it is as though alice and bob had already met 6 seconds before you started observing them.
I don't understand what you exactly want to imply by "deriving an equation from the first graph" but simple newtonian equations (like equations of motion) often give negative solutions of time (which are usually discarded). The most common example is in the second equation of motion (s = ut + 1/2at²) which is a quadratic in t.

[nimble storm]

Thanks! That explains a lot