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.

 

[blue_raver22]

Textbook authors typically get the formulas for position and motion functions from the principles of calculus and kinematics. These principles involve the use of derivatives and integrals to describe the position, velocity, and acceleration of a moving particle. The formulas used in solving derivatives, such as the examples provided (s=t^3-6t^2-9t and f(t)=t^2-10t+12), are derived from these principles and are based on the fundamental laws of motion and the relationships between position, velocity, and acceleration.

To derive these types of functions, one would start by defining the variables involved, such as time (t), position (s), velocity (v), and acceleration (a). Then, using the principles of calculus, one would apply the appropriate formulas to solve for the desired function. For example, to derive the function s=t^3-6t^2-9t, one would first take the derivative of this function with respect to time, which would give the velocity function, v=3t^2-12t-9. Then, taking the derivative of the velocity function would give the acceleration function, a=6t-12. Integrating the acceleration function would then give the original position function, s=t^3-6t^2-9t.

In summary, the formulas used for position and motion functions are derived from the principles of calculus and kinematics, and textbook authors use these principles to provide examples to help students understand and apply these concepts. It is important to understand the underlying principles and relationships between position, velocity, and acceleration in order to derive these functions accurately.