If I am asked to devise an experiment to find an expression for the acceleration of a uniform sphere (ball bearing) rolling down an inclined plane, as a function of the angle of incline. Hence deduce the value of $g$ within 1%.
How would I do this accurately using simple equipment such as rulers and stopwatches? I.e without motion capture or breaker circuits? I think this method will suffice, but is there a way to account for the error more precisely?
I gather that the large source of error you are worried about is the ability of the experimenter to accurately hit the start/stop button on the stopwatch at the start/stop of the ball's journey down the ramp.
Before I directly answer your question, let's estimate how bad the experimental error will be (reasonably close, without actually doing the experiment).
Typical human reaction time is 2-3 tenths of a second, somewhat increasing with age. I would expect the error to be smaller at the top of the ramp (easier to synchronize with "3..2..1..go!"). I would also expect the experimenter to be able to somewhat anticipate when the ball will reach the end of the ramp (because they can see the ball rolling towards it), and these timings may even somewhat cancel each other out. So for argument's sake (not to be used as an actual estimate; that's what science is for), let's be optimistic and say the total amount of reaction error would be around 100ms. For a typical length of ramp used in such an experiment, ball bearing travel times would be on the order of 1 second (1000ms).
Thus, your reaction error is:
$\frac{val_{exp} - val_{known}}{val_{known}} = \frac{(1000+100)ms - 1000ms}{1000ms} = 10\%$
Note: When applied to a calculation of g, the above error will actually be worse, on the order of $g \pm 17\%$, left as an exercise to the reader to verify.
So, yes, a 10% error is far from your desired 1%. Can we do better?
You'd think you could just use a longer ramp or a shallower incline. Unfortunately, to get < 1% error with a 100ms (total) error, you need a 10 second roll, the length of ramp required with a shallow ($10^{\circ}$) incline is given by:
$s = v_{i}t + \frac{1}2at^2$ but since $v_{i} = 0$,
$s = \frac{1}2at^2$
And we know $t = 10 sec$ and $a = g * \sin \theta = 9.81m/s^2 * \sin 10^{\circ} = 1.703m/s^2$
$s = 85.15 m$
I imagine that ramp length would have several practical limitations.
Unfortunately, not much. Given the constraints in the question regarding "ordinary" measurements with rulers and stopwatches, a typical classroom experiment of this type will have a fairly large error. Using many trials and some statistical analysis (if this is a senior high school or university-level experiment) to get a confidence interval for your experimental value might improve the analysis, but not the actual error. Put another way, no math is capable of going back in time and pushing that stopwatch button for the experimenter. Plus, you asked how to improve the experiment, not the analysis.
However, as a teaching tool, that error is actually a rather valuable part of the experiment. It's one of those rare experiments where the value you're trying to measure (g) is very well known, and the sources of error are at the same time significant, but also easy to imagine, and not too difficult to estimate.
Remember, science isn't just about results. It's about questions. (Right, well, tell that to my grant application reviewers.)
