Calculating an equations error given the error of a single value

[buttermellow]

Say you had an equation like E = .5mv². If in an experiment the velocity is measured, but has an error of 10%, what would be the total error in calculating E?

 

[I like Serena]

Hi buttermellow, welcome to PF!

Is this a homework question?

If so, what do you think it should be?
That way we can help you better to understand how to calculate such errors...

 

[ZealScience]

What about m? Just use addition of percentage error here. While making product percentage errors are added, while making sums just add errors. After all, uncertainty of E depends on m as well.

 

[HallsofIvy]

There is an engineers rule of thumb that says that when measurements are added or subtracted their errors add. When measurements are multiplied or divided, their relative errors (error divided by the value) add.

If f= xy and x has error dx, y has error dy, then f could be as large as (x+ dx)(y+ dy)= xy+ xdy+ ydx+ dxdy. Neglecting the small dxdy (if dx and dy are small, dxdy will be much smaller), the error is xdy+ ydx so the relative error is (xdy+ ydx)/(xy)= dy/y+ dx/x.

With 0.5mv^2, for small errors dm and dv, the relative error is dm/m+ 2dv/v.

 

[PJC01]

The total error in calculating E would depend on the type of error that is present in the velocity measurement. If the error is a random error, meaning it is equally likely to be positive or negative, then the total error in calculating E would be 10% of the calculated value of E. This is because the error in the velocity measurement would be propagated through the equation and result in a 10% error in the final calculated value of E.

However, if the error in the velocity measurement is a systematic error, meaning it consistently overestimates or underestimates the true value, then the total error in calculating E may be larger or smaller than 10%. This is because the systematic error would be consistently propagated through the equation, resulting in a larger or smaller error in the final calculated value of E.

To accurately determine the total error in calculating E, it is important to identify and quantify the type of error present in the velocity measurement. This can be done through careful calibration and validation of the measurement equipment, as well as conducting multiple trials and analyzing the data for any trends or patterns that may indicate a systematic error. Overall, it is crucial to consider and account for all sources of error in scientific experiments in order to ensure the accuracy and reliability of our calculations and results.