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Calculating uncertainty

  • Thread starter jgray
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  • #1
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Homework Statement



An object of mass m = 2.3±0.1 kg is moving at a speed of v = 1.25±0.03 m/s. Calculate the kinetic energy (K = 1 mv2) of the object. What is the uncertainty
in K?

Homework Equations



k=1/2mv^2

The Attempt at a Solution


I have figured out that the kinetic energy is 1.8 J, but how do I figure out the level of uncertainty for this question? We do not use derivatives yet.
Can I take the equation for uncertainty of a power and uncertainy of a constant and add them together? :
change in z= k change in x
=1/2 * 0.1kg

change in z= nx ^n-1 * change in x
=2 * 1.25 ^2-1 * 0.03

then add them together to give an uncertainty of + or - 0.125?
 

Answers and Replies

  • #2
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When multiplying/dividing uncertainties, you just add the ratios in quadrature. But in my opinion it is easiest to just do
σ[itex]_{k}[/itex][itex]^{2}[/itex] = (∂[itex]_{k}[/itex]/∂[itex]_{m}[/itex])[itex]^{2}[/itex] * σ[itex]_{m}[/itex][itex]^{2}[/itex] + (∂[itex]_{k}[/itex]/∂[itex]_{v}[/itex])[itex]^{2}[/itex] * σ[itex]_{v}[/itex][itex]^{2}[/itex]
when you learn uncertainties a little more in depth, I think you will find it is much easier to use that with larger expressions.
 
Last edited:
  • #3
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sorry I don't really know what that means though
 
  • #5
haruspex
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When multiplying/dividing uncertainties, you just add the ratios in quadrature. But in my opinion it is easiest to just do
σ[itex]_{k}[/itex][itex]^{2}[/itex] = (∂[itex]_{k}[/itex]/∂[itex]_{m}[/itex])[itex]^{2}[/itex] * σ[itex]_{m}[/itex][itex]^{2}[/itex] + (∂[itex]_{k}[/itex]/∂[itex]_{v}[/itex])[itex]^{2}[/itex] * σ[itex]_{v}[/itex][itex]^{2}[/itex]
when you learn uncertainties a little more in depth, I think you will find it is much easier to use that with larger expressions.
That's fine when uncertainties are given in terms of standard deviations. It might not be appropriate when given in terms of ±.
If the lengths of two components to be manufactured have specs of ±1mm, and they are to be joined end to end, then the uncertainty in the total length is ±2mm. An engineer relying on the total uncertainty being only ±√2mm would soon be out of a job.
A key issue is what is the definition of 'uncertainty' here. If it means standard deviation then you first have to convert the ± data to a standard deviation, and for that you need to know the distribution of the error. In particular, consider the case of measurements taken by eye against a graduated scale. The measurer will round to the nearest unit on the scale. The error therefore has a uniform distribution, ± half the scale unit size. The sum of two such measurements has a different distribution.
jgray, unless you have been taught to use Panphobia's formula for such questions, I suggest just considering the extreme values for the energy that can arise from the ranges of possible values for mass and velocity.
 

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