The Corrections (to '74 UALR lab report)

What I’m doing here is correcting some of my old mistakes.  In my first-ever physics lab report (see “Car Speed Measurements UALR ’74"), I mistakenly rewrote 4/100 as .004 instead of 0.04, and I used .004 to calculate the error or uncertainty in the measured speed of a car.  Well, of six cars.  So I came out with an underestimate of the error in the measurements of speed.

 

As discussed in the lab handout, the formula for “error propagation” in this case is

 σ_v/v  =  σ_t/t  +  σ_d/d,

where v is the velocity, calculated earlier from the measured time, t, and measured distance, d.  The little sigmas represent the calculated or estimated uncertainties in the speed, distance and time.  We use t, d, and v and our estimates of σ_t and σ_d to find σ_v.  This formula simply says that the relative error in speed is the sum of the relative errors in time and distance.

The values for σ_t and σ_d are, respectively, 0.42 sec and 4 meters.  The distance d is always 100 meters, measured along the roadside before the experiment began.  One person estimates when the rear bumper of a car passes the starting point, signaling to the person with the stopwatch standing at the 100 meter point to start timing.  Then the 100 meter person stops timing when the rear bumper passes the 100 meter point.  That gives t.  Another person writes down the license number of the car.  The speed is then 100 meters divided by the measured number of seconds.

So the first thing is to check my calculated speeds. They’re all right! The speed uncertainties can now be recalculated.

Case 1.  σ_v  =  (σ_t/t  +  σ_d/d)v  =  (0.42s/5.5s  +  0.04)(18.2 m/s)   =  2.12 m/s 

Case 2.   …      (0.42/6.2  +  0.04) (16.1)  =  1.73 m/s

 

Case 3.  …    (0.42/7.2  +  0.04) (13.9)  =  1.37 m/s

Case 4.  …    (0.42/4.8  +  0.04)(20.8)  =  2.65 m/s

Case 5.  …    (0.42/6.0  +  0.04)(16.7)  =  1.84 m/s

Case 6.  …  (0.42/7.0  +  0.04)(14.3)  =  1.43 m/s

By license number, the speeds and their uncertainties are:

CCH 255                     18.2  ±  2.12 m/s                    40.8  ±  4.75 mph

CIL 164                       16.1  ±  1.73 m/s                    36.1  ±  3.88 mph

AAW 197                   13.9  ±  1.37 m/s                    31.1  ±  3.07 mph

DSY 611                     20.8  ±   2.65 m/s                   46.6  ±  5.94 mph

AAV 637                    16.7  ±  1.84 m/s                    37.4  ±  4.12 mph

MWM 646 (TX)        14.3  ±  1.43 m/s                    32.0  ±  1.43 mph

The Texan wasn’t living up to his or her reputation (in Arkansas) of being a speeder.  Could’ve been an Arkie driving a Texas car, or many other possibilities. 

Oh, you can give marriage a whirl

If you’ve got some cash in your purse

But don’t marry no one but a Texas girl

‘Cause no matter what happens, she’s seen worse

--a “courting song” introduced and sung by Pete Seeger (he was strumming a banjo also) on a folk music set of albums I have, given to my brother Steven and me by our dad in 1974.