# Calculating uncertainty

1. Jan 10, 2014

### jgray

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

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?

2. Relevant equations

k=1/2mv^2

3. 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?

2. Jan 10, 2014

### Panphobia

When multiplying/dividing uncertainties, you just add the ratios in quadrature. But in my opinion it is easiest to just do
σ$_{k}$$^{2}$ = (∂$_{k}$/∂$_{m}$)$^{2}$ * σ$_{m}$$^{2}$ + (∂$_{k}$/∂$_{v}$)$^{2}$ * σ$_{v}$$^{2}$
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: Jan 10, 2014
3. Jan 10, 2014

### jgray

sorry I don't really know what that means though

4. Jan 10, 2014

### Panphobia

5. Jan 10, 2014

### haruspex

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.