Error Propagation

asleight

1. The problem statement, all variables and given/known data

Given that a puck's velocity is speed [tex]v[/tex] at an angle [tex]\theta[/tex] (measured in radians) with the x-axis, we know that the puck's x-velocity is [tex]v\cos(\theta)[/tex]. Given the error in [tex]v[/tex] is [tex]\sigma_v[/tex] and the error in [tex]\theta[/tex] is [tex]\sigma_\theta[/tex], what is the resulting error in the puck's x-velocity?


3. The attempt at a solution

Solving for partials, we get:

[tex]\sigma_{v_{x}}=\sqrt{\left(\cos(\theta)\sigma_{v}\right)^2+\left(-v\sin(\theta)\sigma_{\theta}\right)^2}[/tex].

Or, using proportionalities of errors, we find:

[tex]\sigma_{v_{x}}=\sqrt{\left(\frac{\sigma_{v}}{v}\right)^2{v_{x}}^2+\left (\frac{\sigma_{\theta}}{\theta}\right)^2{v_{x}}^2}[/tex].

These yield two different values... Which is a real propagation?

LowlyPion

This link touches on dealing with error propagation for angles.
http://instructor.physics.lsa.umich....rors/prop.html

Since you are interested in the product of two measured values that would suggest that your second method would be the final step.

But arriving at the fractional uncertainty of the Trig function suggests finding the absolute uncertainty in the function first.

By the Rule 4 at the link I cited above you might model that as σ_f = dF(θ)/dθ = σ_θSinθ

From that calculate the relative uncertainty as σ_θSinθ/Cosθ = σ_θTanθ ?

By my method I think that would make it

σ_vx = ((σ_v/v)² + (σ_θTanθ)²)^1/2

asleight

Thank you, Lowly, but I guess I'm still confused. All year we've used the RSS of the partials to show the error in the unknown parameter but, recently, my TA introduced the idea of proportionality of squares. That is, [tex](dC/C)^2 = (dA/A)^2+(dB/B)^2[/tex]. Which is right?

LowlyPion

By my method I think that would make it

σ_vx = ((σ_v/v)² + (σ_θTanθ)²)^1/2