Let’s say we have a scenario of a ball being released from the top of the building. This can be modeled simply with the kinematics equation $S=ut +\frac{1}{2}at^2$, which reduced to $S=\frac{1}{2}at^2$. We are given $\Delta t, t, \Delta S, S$, are we are to find $a, \Delta a$.
Firstly, I have no problems calculating the absolute portion of the uncertainty.
Here is my problem: Differentiating $S=\frac{1}{2}at^2$ gives me $\frac{\Delta S}{S}=\frac{\Delta a}{a}+2\frac{\Delta t}{t}$. However, substituting these values gives me a wrong value of $\Delta a$.
The correct approach should have been to rearrange the equation to $a=\frac{2S}{t^2}$, and then solve $\frac{\Delta a}{a}=\frac{\Delta S}{S}+2\frac{\Delta t}{t}$. As can be seen, there appears a contradiction.
Further substitution of $S=82m,\Delta S=1m,t=4.1s, \Delta t=0.2s$ to solve for $a, \Delta a$ using the second equation and then putting this value back into the first gives me a contradiction.
I would like to know which one is correct and which should be used because both seem correct to me.
I have discovered that the addition/subtraction of uncertainties is as follows. Let’s say $(A\pm\Delta A)+(B\pm\Delta B)=(C\pm\Delta C)$.
Then $C_{max}=(A+\Delta A)+(B+\Delta B), C_{min}=(A-\Delta A)+(B-\Delta B)$. Referring back to the definition of uncertainty, $C+\Delta C$ is the average of the minimum and maximum of $C$, thus giving us $C=A+B$ and $\Delta C=\Delta A+\Delta B$.
Using this principle, I am however confused by what I get. $C_{max}=(A+\Delta A)(B+\Delta B), C_{min}=(A-\Delta A)(B-\Delta B)$. Expanding, I got $C=AB +\Delta A\Delta B$, which was contradictory to what I have learnt. I got $\Delta C=A\Delta B + B\Delta A$, which was correct though... This raises a new problem, as I am now unsure as to why the rule applies to multiplication.