How Does Parametrization Help Describe Particle Motion in Mathematics?

[AAMAIK]

TL;DR

What is the purpose of parametrization
What is a parameter?

To describe the equation of a line, in 2 dimensions, we need a (point on the line + slope to measure slantiness) or two points. Another way: The trajectory of a moving point along the line. Suppose that the moving point initially is at a point of know coordinates r0=(x(t=0), y(t=0), z(t=0)) and that it moves at a constant speed on the line. From the above, the velocity vector has constant components v=(a1,a2,a3) independent of time, and the particle moves along one direction on either side of the initial point. The position of the moving point at any time , t
r=r0+Δr=r0+vΔt=>
x(t)=x(t=0)+a1*t
y(t)=y(t=0)+a2*t
z(t)=z(t=0)+a3*t
If the particle moves at variable speed
x(t)=x(t=0)+ ∫a1(t)dt
y(t)=y(t=0)+ ∫a2(t)dt
z(t)=z(t=0)+ ∫a3(t)dt
With this approach, I can describe the position of any point along the line. But I have problems with this approach, one, if given the x=2 I want to find the corresponding y and z such that the point (x=2, y,z) lies on the line. What if the trajectory of the moving particle does not go through x=2.
Secondly, this approach was based on the fact that the trajectory is along the line (given), it was made up. I don't understand parametrization, I view it as describing things that could be deduced easily from the givens of a moving point not the arbitrary curve in the plane or space it was introduced to serve to describe in the first place.

 

[BvU]

What if the trajectory of the moving particle does not go through x=2

Then you don't find a solution

You could consider a parametrization as a kind of coordinate transform. In that sense $$x=\sin\omega t\\y = \cos\omega t$$ is also a parametrization

 

[AAMAIK]

I wasn't trying to find a solution to the problem, I was illustrating why we need parametrization in the first place to describe trajectories in space or planes through a simple example of a line.

 

[Stephen Tashi]

Summary:: What is the purpose of parametrization
What is a parameter?

But I have problems with this approach, one, if given the x=2 I want to find the corresponding y and z such that the point (x=2, y,z) lies on the line. What if the trajectory of the moving particle does not go through x=2.

If the the particle does not pass through x = 2 there should be no solution for t to the equation x(t) = 2. If There is a solution t0 to this equation, the particle passes through (2, y(t0),z(t0)).

Secondly, this approach was based on the fact that the trajectory is along the line (given), it was made up. I don't understand parametrization, I view it as describing things that could be deduced easily from the givens of a moving point not the arbitrary curve in the plane or space it was introduced to serve to describe in the first place.

That's a interesting thought. Of course you can't find the velocity or acceleration of point moving along a curve (such as a line) only from knowing the locus of points on the curve. Two particles can follow the same curve at different rates, just as two cars can travel the same road at different rates. So to have adequate "givens" to find the velocity and acceleration of a particle, you must know the rate of travel as a function of something. If we think of a curve parameterized by a variable representing time t, it is straightforward to find the rates of travel in the (x,y,z) directions by differentiation. I can imagine a situation where, we are given the locus of points (x,y,z) on the trajectory and know the particles velocity as a function of those locations. So our velocity information has the format ( vx(x,y,z), vy(x,y,z), vz(x,y,z)). How would we find the acceleration of the particle from that information? Is this a situation where acceleration can be deduced easily?