Error propagation formulas δx represents the uncertainty of the physical quantity x, so that we should write (x ± δx). Given three measurements a ± δa, b ± δb, c ± δc, this is how you should propagate their errors (δa, δb, δc) to obtain the uncertainty of the final result for each operation. Let’s call the final result R, and its associated final uncertainty will therefore be δR. Ultimately you want to find δR, to finally express your final answer as R ± δR. Sum/subtraction: R = a + b – c or R = a + b + c or R = c – a – b etc 𝛿𝑅 = 𝛿𝑎! + 𝛿𝑏 ! + 𝛿𝑐 ! Product/division: R = a/bc or R = abc or R = c/ab etc 𝛿𝑅 = 𝑅
𝛿𝑎
𝑎
!
𝛿𝑏
+
𝑏
!
𝛿𝑐 !
+
𝑐
Involving a constant: R = (constant) ab/c 𝛿𝑎 !
𝛿𝑏 !
𝛿𝑐 !
𝛿𝑅 = 𝑅
+
+
𝑎
𝑏
𝑐
still the same formula as before…. Example of an application: Say you need to calculate the final momentum P = p1 + p2 (this equation is a sum), where p1 =
m1v1 and p2 = m2v2 (these equations involve products).
You have performed an experiment where you were able to measure m1 ± δm1, m2 ± δm2, v1 ±
δv1, and v2 ± δv2.
The ultimate goal is to get P ± δP. How can you do this? You must do it in parts. First get δp1
and δp2, and then move on to calculate δP. Here we go:
𝛿𝑝! = 𝑝!
Now:
!!! !
!!
+
!!! !
!!
and
𝛿𝑃 = (𝛿𝑝! )! + (𝛿𝑝! )!
Present your final answer as P ± δP.
𝛿𝑝! = 𝑝!
!!! !
!!
+
!!! !
!!