# Combining Multiple Rules for Error Propagation

• ELLE_AW
In summary, when calculating the kinetic energy of an object with mass m=2.3±0.1kg moving at a speed of v=1.25±0.03m/s, the uncertainty of the kinetic energy can be estimated using the product rule for correlated variables or added in quadrature for uncorrelated variables. The kinetic energy is K = 1/2mv2 = 1.8 J, and the uncertainty is δK = 0.17.
ELLE_AW

## Homework Statement

1. An object of mass m=2.3±0.1kg moves at a speed of v=1.25±0.03m/s. Calculate the kinetic energy (K=1/2mv2) of the object and estimate the uncertainty δK?

## Homework Equations

- Addition error propagation--> z = x + y and the Limit error--> δz = δx + δy

- Exponent error propagation --> z = xn and the Limit Error --> δz = nxn-1(δx)

- K = 1/2mv2

## The Attempt at a Solution

This is what I attempted, but I really don't think it's right. I basically just included the exponent error propagation, but how does the multiplication of mv2 get incorporated?

- K = ½ mv2 = ½ (2.3kg)(1.25m/s)2 = 1.7969 kg m2 s-2 = 1.8 J

- Uncertainty of K = (m)2v1(δv) = (2.3)(2)(1.25)(0.03) = 0.1725 = 0.17
How do I combine these two rules when calculating the uncertainty of the kinetic energy?

ELLE_AW said:
Relevant equations
You seem to be missing the product rule:
If z=xy then δz/z=δx/x+δy/y

ELLE_AW said:
Addition error propagation--> z = x + y and the Limit error--> δz = δx + δy
This is error propagation for correlated variables x and y. If your variables are uncorrelated, they should be added in quadrature, i.e.,
$$\delta z^2 = \delta x^2 + \delta y^2.$$
The same is true for uncorrelated relative errors in the case of a product.

Got it, thank you!

## 1. What is error propagation and why is it important in scientific research?

Error propagation refers to the process of calculating and predicting the uncertainties or errors in a measurement or calculation based on the uncertainties of the input variables. It is important in scientific research because it allows for a more accurate and precise understanding of the data and results, helping to determine the reliability and significance of the findings.

## 2. How do you combine multiple rules for error propagation?

To combine multiple rules for error propagation, you need to first identify the individual rules that apply to each variable in your calculation. Then, you can use the general method of error propagation, which involves finding the partial derivatives of each variable with respect to the others and using them to calculate the overall uncertainty.

## 3. Can error propagation be used for any type of measurement or calculation?

Yes, error propagation can be used for any type of measurement or calculation, as long as the uncertainties of the input variables are known. It is a versatile method that can be applied to a wide range of scientific disciplines, from physics and chemistry to biology and engineering.

## 4. How does combining multiple rules for error propagation affect the overall uncertainty of a measurement or calculation?

Combining multiple rules for error propagation can either increase or decrease the overall uncertainty of a measurement or calculation, depending on the specific rules being used and the magnitude of the uncertainties. In some cases, the combined uncertainty may be smaller than the individual uncertainties, resulting in a more precise measurement or calculation.

## 5. Are there any limitations or assumptions when using multiple rules for error propagation?

Yes, there are some limitations and assumptions when using multiple rules for error propagation. One limitation is that it assumes the uncertainties of the input variables are independent and uncorrelated. Another assumption is that the uncertainties are normally distributed. Additionally, the method may not be accurate for highly nonlinear calculations or when the uncertainties are very large compared to the values being measured or calculated.

• Introductory Physics Homework Help
Replies
5
Views
3K
• Introductory Physics Homework Help
Replies
12
Views
29K
• Introductory Physics Homework Help
Replies
4
Views
3K
• Introductory Physics Homework Help
Replies
2
Views
3K