Velocity Error Measurement by Linear Approximation

[Bluewolf1986]

I am stuck on the following problem which is from the Linear Approximation: Measurement Errors section of my Calculus 1A: Differential Calculus Class:

Suppose that you measure the velocity of an object by measuring that it takes 1 second to travel 1.2 meters. The measurement error is .001 meters in distance, and the error in time is .01 second. What is the absolute value of the error in the linear approximation for the velocity?

(Enter decimal answer to three decimal digits. Enter the units of the error using m to denote meters and s to denote seconds.)

So far I have identified the velocity (V) which is 1.2 m/s and the change in distance (.001 m) and the change in time (.01 s). I tried to find the absolute value in the
measurement error by dividing the change in distance over the change in time (0.10 m/s), adding it to the 1.2 m/s measured velocity, then subtracting that value by 1.2 m/s (1.2 m/s -1.3 m/s) and ended up with -0.10 m/s which obviously has the absolute value of 0.10 m/s. This answer was incorrect, however. I then realized that V + (Delta) V should give the correct answer, however I am confused as to how to find these values using a linear approximation formula for constants. Please help by providing a hint as to what formulas to use to find this absolute error value through linear approximation. Thank you.

 

[HallsofIvy]

Yes, since you measure the distance as 1.2 m with an error of 0.001 m, the distance could be as long as 1.201 m or as little as 1.199 m. Since you measure the time as 1 s with an error of 0.01 seconds the time could be as great as 1.01 s or as little as 0.09 s.. The largest possible speed would be if you went 1.201 in 0.09 s, 1.201/0.09= 13.34 m/s. The least possible speed would be if you went 1.199 m in 1.01 s, 1.199/1.01= 1.187 m/s. The absolute error is 13.34-1.187= 12.502 m/s.

 

[Deleted member 4993]

Another way to consider error propagation would be to look at the % error:

\(\displaystyle \displaystyle{velocity \ = \ \dfrac{distance*(1\ \pm \delta d)}{time*(1\ \pm \delta t)} }\)

\(\displaystyle \displaystyle{velocity \ = \ \dfrac{distance}{time} \ \ * \ \ (1 + \delta d - \delta t)}\)....... considering only "+" from above

and so on.....