Physics A Level
P1 Practical skills at AS Level P1.10 Combining uncertainties
Physics A Level
P1 Practical skills at AS Level P1.10 Combining uncertainties
When quantities are combined, for example, multiplied or divided, what is the uncertainty in the final result?
Suppose that quantity $A = 1.0 \pm 0.1$ and that $B = 2.0 \pm 0.2$, so that the value of $A + B$ is 3.0. The maximum likely value of $A + B$, taking into account the uncertainties, is 3.3 and the minimum likely value is 2.7. You can see that the combined uncertainty is $ \pm 0.3$, so $A + B = 3.0 \pm 0.3$. Similarly, $B - A = 1.0 \pm 0.3$
When quantities are added or subtracted, their absolute uncertainties are added. A simple example is measuring the length of a stick using a millimetre scale. There is likely to be an uncertainty $0.5 mm$ at both ends, giving a total uncertainty of $1.0 mm$.
When quantities are multiplied or divided, combining uncertainties is a little more complex. To find the combined uncertainty in this case, we add the percentage uncertainties of the two quantities to find the total percentage uncertainty.
Remember, you always add uncertainties; never subtract.
Where quantities are:
- added or subtracted, then add absolute uncertainties
- multiplied or divided, then add percentage or fractional uncertainties.
Questions
17) You measure the following quantities:
$A = (1.0 \pm 0.4)m$
$B = (2.0 \pm 0.2)m$
$C = (2.0 \pm 0.5)m\,{s^{ - 1}}$
$D = (0.20 \pm 0.01)s$
Calculate the result and its uncertainty for each of the following expressions. You may express your uncertainty either as an absolute value or as a percentage.
a: $A + B$
b: $B − A$
c: $C \times D$
e: ${A^2}$
f: $2 \times A$
g: the square root of $(A \times B)$. (Recall that the square root of x can be written as ${X^{\frac{1}{2}}}$.)
18) A rifle bullet is photographed in flight using two flashes of light separated by a time interval of $(1.00 \pm 0.02)ms$. The first image of the bullet on the photograph appears to be at a position of $(22.5 \pm 0.5)cm$ on a scale underneath the flight path. The position of the second image is $(37.5 \pm 0.7)cm$ on the same scale. Find the speed of the bullet and its absolute uncertainty.