Function notation and shifting functions

[PFuser1232]

Suppose two people, X and Y, have two different stopwatches. X starts his/her stopwatch as some particle passes an origin. We can model the velocity of the particle by $\vec{v}(T)$, where $T$ is the reading on the first stopwatch. After an amount of time $\Delta t$, Y starts his/her stopwatch ($T = t + \Delta t$). Is it correct to model the velocity of the particle as $\vec{v}(t)$ where $t$ is the reading on the second stopwatch? Or should we change the letter used to represent the function [to $\vec{u}(t)$ for example]?

 

[DrClaude]

It depends. If you are always using $\vec{v}(T)$ and $\vec{v}(t)$, then there is no ambiguity. But if somewhere you are to write something like $\vec{v}(0)$, then it is not clear which time you are referring to.

 

[PFuser1232]

It depends. If you are always using $\vec{v}(T)$ and $\vec{v}(t)$, then there is no ambiguity. But if somewhere you are to write something like $\vec{v}(0)$, then it is not clear which time you are referring to.

I think it's because the argument of the function $\vec{v}$ as defined earlier always represents the reading on the first stopwatch. To represent velocity in terms of the reading on the second stopwatch a new function is forced upon us such that $\vec{u}(t) = \vec{v}(T)$ for all $t$. Is this correct?

 

[DrClaude]

I think it's because the argument of the function $\vec{v}$ as defined earlier always represents the reading on the first stopwatch. To represent velocity in terms of the reading on the second stopwatch a new function is forced upon us such that $\vec{u}(t) = \vec{v}(T)$ for all $t$. Is this correct?

That equation looks correct. Since $T = t + \Delta t$, you recover the correct behavior, e.g.,
$$
\vec{u}(0) = \vec{v}(\Delta t)
$$