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i have a very basic question from school days. what does it mean to say an object is moving with uniform speed? it seems to me now that it should be an unit dependent concept. for example if speed is the derivative of distance traveled, i.e. $X'(t)$ , and I decide to measure distance on a new scale $F(X)$, a monotonic function of $X$, but not a linear multiple. Then, speed in that scale at a point t would be $F'(X(t))$. $X'(t)$ , which will not be constant as $F$ can be an arbitrary increasing function.

and if indeed "uniformness of speed" is a unit dependent concept, what does it mean to say, light travels at a constant speed ? also this would mean "uniform acceleration" is also a unit dependent concept. how is then, "acceleration due to gravity" a universal constant ? would it cease to be a constant if I measured acceleration in $\log(m/s^2)$ ?

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Neat trick.

Fortunately, you're not actually changing anything physical. The graph drawn in those units would not be a straight line but the particle would still move the same distance in any fixed amount of time, i.e. it would still be uniform velocity. So it might move 8 units of distance in a 1 second interval and 16 units in the next 1 second interval but the actual distance moved would be the same in both intervals of time.

What you're doing is more like a change of coordinates than a change of units.

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  • $\begingroup$ can you clarify with an example ? suppose at time points 1, 2 and 3, an object is at 5, 10 and 15 metres from the origin. measured in log this would be log(5), log(10) and log(15). do you agree that distance travelled is log(10)-log(5)=0.69 and log(15)-log(10)=0.40 which are not equal. Supposed "metres" did not exist and you only knew about log-meters. In what sense would you say "it moves equal distances in equal time" ? $\endgroup$
    – S B
    Commented Jul 1, 2011 at 2:18
  • $\begingroup$ @S B, I would say that if I took a stick and laid it off between log(10) and log(5) and log(15) and log(10), I would be able to lay it down the same amount of times between each of them. This is the basic operative definition of distance. $\endgroup$
    – knucklebumpler
    Commented Jul 1, 2011 at 2:26
  • $\begingroup$ @knucklebumper: sorry still confused. if the length of the stick is L. for what you say to be true, both these lengths have to be "r * L" for some r (r is the possibly fractional number of times you repeat the stick). But that means, both lengths have to be equal, but they are unequal (one 0.69 the other 0.40). $\endgroup$
    – S B
    Commented Jul 1, 2011 at 2:33
  • $\begingroup$ @knucklebumper i think i understand a little what you mean by "not changing anything physical". so if I measure the distance that it travelled in the first time interval using a stick. then the same stick can be laid in the path traversed in the next time interval. so "assuming" that it had a uniform speed during the first interval, the second interval has the same speed. of course one can not use "distance traveled" by that same object in 1 second (the stick) as the unit for distance, so we need a common reference...continued. $\endgroup$
    – S B
    Commented Jul 1, 2011 at 3:37
  • $\begingroup$ @knucklebumber continued...so we can define the distance traveled by light in one second as the unit of distance. then basically the object has a constant speed in light-year per second. change of unit like log is not meaningful. the length of the stick (used as standard of measure) in light-seconds would remain the same. does this make sense ? $\endgroup$
    – S B
    Commented Jul 1, 2011 at 3:38

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