What is the SI unit for "distance covered in nth second"? Please explain.

SI units are different. Customary are more  like a ZOO:
 
 size of fonts is points, fabric in yards, trips in miles, or nautical
miles, or parsecs, or light years ..
  
  mass of jewelery in carats, mass of wheat, of oil .. in other units ..
 

 
 SI is different, there are 5 (OR 7) BASIC QUANTITIES which have
  basic units,  other units are derived from the basic ones and
  
         There is ONE unit and ONLY ONE basic  unit for EACH QUANTITY


general - intro:

http://physics.nist.gov/cuu/Units/introduction.html

specifics:

for distance. be it 1st OR nth, covered on uncovered , it is:

length meter m

http://physics.nist.gov/cuu/Units/units.html


So, distance is always measured in decimal multiples of meter, 
units such as mm, km or Mm  ...

the prefix m k M..  is selected based on size of quantity

http://en.wikipedia.org/wiki/Physical_unit
http://en.wikipedia.org/wiki/Physical_quantity

 If you need further clarification, do a RFC, but please, 
 include more of what you are wondering about, what the problem is, OK?
 
 Hedgie

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Request for Answer Clarification by dka-ga on 14 Feb 2005 00:24 PST

The displacement comes out as a solution of the differential equation
ds/dt = u + a*t

The time steps, dt, are the basic time unit T.

This *differential equation* is the one which gets discretized, into :
Distance covered in nth second, s(n) = s_(n+1) - s_(n) = (u + a*n*T)*T
(The units, m, match here!)

Next, choose T, a small time step, as a time unit, and measure
everything in terms of it ... just like seconds on a watch. Now, what
does T become in these new units ? It is just '1'.

Won't the unit of "distance covered in nth second" be m? Can you
please clarify why it has to be m/s?

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Clarification of Answer by hedgie-ga on 15 Feb 2005 07:24 PST

dka
      
      We have more problems here then just choice of unit.
      I will try - but you need to  understand for that $2 I can only give
      it that much of time amount of time. 
      So - you may have to have to do some
      reading of references I will provide.
      
      First issue is terminology
      
      quantity      unit         dimension
e.g.    time         s              s           (s = second)
        time        ms              s            (ms - millisecond)
	speed       km/s            m/s         (km = kilometer s second)
	distance     km  or m            m
...
...
	This is explained in references I gave you in the answer.
        You need to click around a bit:
e.g.	   
	   The dimension of a physical quantity is the type of unit needed to
express it...
	  http://en.wikipedia.org/wiki/Dimensional_analysis  
There are many pages on this, some more simple than the above, e.g.	  
	     http://www.chemistrycoach.com/use.htm
	  
In brief:  Each equation, all the time, must be dimensionally correct.
	    
 Your first equation
	     
	     dS/dt = u + a*t
	     
	     is OK.  dS/dt is measured in m/s , so is u, and 
	     
	     a = acceleration {in m/ (s*s) }   and so a*t  is also in m/s  
	     
	     meaning, eq. is dimensionally correct.
	     
	     Now we solve it:
	    
We choose a small time step. THAT IS NOT A UNIT OF TIME, unit is still
s or ms ...   (not important right now)
	     
	THIS "..      and measure
everything in terms of it ... just like seconds on a watch. Now, what
does T become in these new units ? It is just '1'. .." 

  does not make sense. You cannot measure S in units of time ..
  
     t is quantity - time , and unit will be s or ms,
     distance m will be in m (or km)
     speed, u or dS/dt will be in m/s  
   always.  
     Who says that distance S is in m/s ??
     
     
     Here are few equations of the type you have
     http://www.glenbrook.k12.il.us/gbssci/phys/Class/1DKin/U1L6c.html
    http://www.ngsir.netfirms.com/englishhtm/Kinematics.htm
     
     Here is how you solve it:
     http://calculus.sjdccd.cc.ca.us/ODE/7-C-1/7-C-1-h.html
    http://math.fullerton.edu/mathews/n2003/Euler'sMethodMod.html
    
    ..
    
      Basically, from your basic eq. you get
      
      S = S0  + u * t  + .5 * a * t * t 
      
     where S is distance - still measured in meters
     
    http://www.mcasco.com/p1consta.html
    
    This explains the numerical solution process
    
     http://www.pma.caltech.edu/~physlab/assignment-4.pdf
     

Hedgie

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Clarification of Answer by hedgie-ga on 16 Feb 2005 22:39 PST

dka,

I want to add two points, after thinking about your question some more:

1) You may devise a system of unit where small time interval T would
be 'unit of time'
   or even a system when time is dimensionless, so the time steps are
just numbers 1, 2, 3 ,,,
   However, these new systems of units will are not a SI system.
   
   If in new system time would be dimesionless,
    and meter will remain unit of length,
   then speed v and  acceleration a
   will be  measured in meters. 
   
2) distance covered in step n is always (in any system of units)    Vave * T ,
    where Va is average speed in that interval.
    That distance will come in meters, not m/s.
    
     if speed v(n)   is  u + a*n*T       at the begining and
              v(n+1) is  V(n) + a * T    at the end of the interval then
	      Vave= V(n) + .5 * a * T     (that .5 is important).
	    
    in SI that speed has dimension m/s  and T is in s , so distance is in m
    in the unusual system, where t is dimensionles, both distance and
speed are in meters
    
Does that makes sense to you?

    Hedgie

SI defines standards, for example, distance / time / weight...  etc.

They do not have a special unit for speed/velocity (distance/time),
but they do have standard unit for distance and time.

Standard unit for distance is in meters (m), and standard unit for
time is in seconds (s), so answer to your question would be,

meters / seconds  (m/s)

The displacement comes out as a solution of the differential equation
ds/dt = u + a*t

The time steps, dt, are the basic time unit T.

This *differential equation* is the one which gets discretized, into :
Distance covered in nth second, s(n) = s_(n+1) - s_(n) = (u + a*n*T)*T
(The units, m, match here!)

Next, choose T, a small time step, as a time unit, and measure
everything in terms of it ... just like seconds on a watch. Now, what
does T become in these new units ? It is just '1'.

Won't the unit of "distance covered in nth second" be m? Can you
please clarify why it has to be m/s?