I currently need to model a dot moving along an arbitrary curve given it's velocity, initial point, and $y=f(x)$ form of equation. I vaguely remember from my high school teaching that it will possibly require using differentiation, integration, and figuring out parametric equation of the curve. What I currently do not understand specifically is that how do you incorporate velocity into parametric function. In the examples I find they can just change x value to t and plug it into the $y=f(x)$ function. However intuition tells me that I should not just do it, and change the value I plug in according to the velocity of the dot.
In other words, what would the difference of parametric equation look like if the dot is moving along a circle to left at velocity of $1$m/s and the other moving at $5$m/s towards the same direction.
p.s.Also would this be assuming that the dot with same speed on two different roads of the same distance(one s-shaped and the other straight line, both $100$ m) would they reach the end at the same time?