Deriving Position & Motion Functions of Moving Particles

[nadia123]

Where do textbook authors get the formula for the position function or the function of a moving particle that are used as examples in solving derivatives? Here are examples:

$$s=t^3-6t^2-9t$$

$$f(t)=t^2-10t+12$$


I am interested on how to derive these kind of functions. How?

 

[radou]

What exactly are you asking? If you're talking about Newtonian physics, all the relations which describe the position of a particle are derived from Newton's law of motion, which is a differential equation.

 

[nadia123]

I am interested to know how are those equations or functions were derived. Yes, these are equations in physics.

I wonder if they were made up by calculus textbook authors and were not based on Newtonian physics because sometimes the functions are not the same. For example:

Sometimes the position function is like this

$$s = 16t^2 - t$$

then in the next example it would look like this

$$s = 16t^3-t^2-t-12$$

 

[MustangGT]

I have never used LaTex before, so excuse me if this looks funny:

It depends on the situation. The general one dimensional equation for linear position is
x(t) = x₀ + v₀t + (1/2)a₀t² + (1/3)jt³

where x₀ is the initial displacement, v₀ is the initial velocity, and a₀ is the initial acceleration. The cubed term represents the change in acceleration with respect to time and is called "jerk." For the reason why some terms are missing in your examples consider:
a particle that is initially at rest would have v₀=0, a particle that starts at the origin would have x₀=0 and so on.