Intrinsic coordinates and an intrinsic description of motion

[fog37]

Hello,

For 2D motion, I understand that velocity, position and acceleration of a point object can be described using the fixed basis vector $\hat {i}$ and $\hat {j}$ and the rectangular coordinates $x(t)$ and $y(t)$ which which are functions of time $t$.

Another option is to use the tangential (to the trajectory) unit vector $\hat {e}_t$ and the normal (to the trajectory) unit vector $\hat {e}_n$. Are the components of the tangential and normal unit vectors supposed to be functions of time $t$ or functions of the scalar arc-length $s$?

Also, why are the tangential and normal unit basis vectors called intrinsic? When is the motion description with these two basis vectors more useful than a description involving the traditional rectangular coordinates?

Thanks!

 

[Cryo]

Too many questions :-)

First: time $t$, and arc-length $s$. They are related, aren't they? i.e. $s=s\left(t\right)$ is defined, and, as long as time-travel is not allowed, $t=t\left(s\right)$ is also defined. So there is no difference which one you use to describe the motion - your choice.

Second: Tangential and normal vector would be called intrinsic because they do not need a definition of basis vector to span the space, i.e. the tangential and normal vectors are intrinsically defined when you define a smooth curve.

Third: The definition in terms of intrinsic vectors may be useful if you consider things from the point of view of observer following the curve, i.e. when you sit on the ship that is going somewhere, and you see a plane flying past, it is easy to say whether the plane's trajectory is parallel or perpendicular to that of the ship.