Example of uncertainty in calculating velocity of ball

You and your partners measure the time it takes for a ball to roll 40 cm and 60 cm. You make 3 trials at each distance.

  trial     time to roll       time to roll 
            40 +/- 0.5 cm      60 +/- 0.5 cm
----------------------------------------------
   1           0.95  s            1.22  s
   2           1.03               1.15
   3           1.01               1.27
----------------------------------------------

 mean          1.00  s            1.21  s

 stdev       +/- 0.04 s        +/- 0.06  s

Here's how to compute the average velocity over this interval of time, and the uncertainty in that average velocity.


      displacement  =    60 cm   -    40 cm    =   20     cm
  
         uncert     =    0.5 cm  +     0.5 cm  =    1.0   cm
      

    time interval   =   1.21 s   -   1.00 s    =    0.21  s
   
         uncert     =    0.04 s  +   0.06 s    =    0.10  s


                        displacement        20 cm
   avg  velocity  =     -------------  =   -------  =  95.2 cm/s
                        time interval       0.21 s
       
 
   uncert in velocity      ( 1.0 cm )     ( 0.10 s ) 
  --------------------  =  ( ------ )  +  ( ------ )  =  0.53
       velocity            ( 20 cm  )     ( 0.21 s )


   uncert in velocity   =  (velocity)  *  0.53  =  50.1 cm/s

Yikes! That's a pretty large uncertainty. Your final result is

       average velocity  =  95  +/- 50    cm/s