# Physics Uncertainty Question?

1. Feb 27, 2012

### RichardT

Physics Uncertainty Question????

1. The problem statement, all variables and given/known data
I have Vy = 0.125tan(54) = 0.17204. I have to find the uncertainty of Vy Where 0.125 = A and tan(54) = B

uncertainty of A (0.125) = 0.0005
uncertainty of B (54 degrees) = 0.5degrees

2. Relevant equations
Uncertainty for Multiplication = uncertainty of Vy/Vy = uncertainty of A/A + uncertainty of B/B

3. The attempt at a solution

Setup: uncertainty of Vy/0.17204 = 0.0005/0.125 + 0.5/tan(54)

My problem is the tangent part: (0.5/tan(54)). do i do that for uncertainty or do i do tan(0.5)/tan(54)??

2. Feb 27, 2012

### Staff: Mentor

Re: Physics Uncertainty Question????

For multiplication the rule is, if Z = X*Y and the independent uncertainties in X and Y are ΔX and ΔY, then
$$\frac{\Delta Z}{Z} = \sqrt{\left(\frac{\Delta X}{X}\right)^2 + \left(\frac{\Delta Y}{Y}\right)^2}$$
A 'cheap and dirty' method for functions of several variables such as your Vy*tan(θ) is to calculate the extremes of the possible results by varying the variable values by their uncertainties and tabulating the results. You can then see how much the results can vary from the central value.

A 'proper' way to do it is to see how the function varies with each variable using partial derivatives. The error contribution of uncertainty Δx of variable x in a function f(x,y,...) is then
$$\Delta f_x = \frac{\partial f}{\partial x} \Delta x$$
and the total uncertainty of f is given by adding all the contributions for each variable in quadrature (i.e. square root of the sum of the squares, just like adding vector components to find a magnitude):
$$\Delta f = \sqrt{\Delta f_x^2 + \Delta f_y^2 + \; ....}$$

EDIT: By the way, I think you'll want to have your Δθ value in radians for the 'proper' method.

Last edited: Feb 27, 2012
3. Feb 27, 2012

### Delphi51

Re: Physics Uncertainty Question????

You must mean that you have V = A*tan(B) and A = .125 ± .0005, B = 54 ± .5.
Neither of those is right!
You could just do the old high school high and low values and take half the difference. Or the error propagation method for tan(B) is to differentiate it:
Z = tan(B)
ΔZ = d(tan B)/dB * ΔB
ΔZ = sec²B * ΔB
You can use the 54 degrees for B, and make sure your calculator is on degrees when you do the secant. But for the ΔB you simply must use the unitless angle - convert the degrees into radians.

Edit: yikes, it must have taken me 20 minutes to sort all that out! I didn't see your post when I started.

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