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.,
$$
\vec{u}(0) = \vec{v}(\Delta t)
$$