1-4 - Units, Estimation and Measurement Limits

Units as a Route Through the Physics

A number on its own is not a physics measurement. $1.20$ could be a distance in metres, a time in seconds, a current in amperes, or a completely different quantity. The unit tells you what has been measured, and often it also tells you how the final quantity was built.

The SI base quantities are the starting set. Their units are not made from other SI units.

A derived unit is an audit trail of that combination.

For example, speed is distance divided by time:

Acceleration is change in velocity divided by time; along one line with no change of direction this also describes the change in speed:

Force is connected to mass and acceleration:

That derived unit is given the special name newton, $\mathrm{N}$, so:

The tempting shortcut is to learn $\mathrm{N}$, $\mathrm{J}$, $\mathrm{Pa}$, $\mathrm{W}$ and other named units as labels. The stronger physics habit is to unpack them when you need to check an equation. If a calculation for speed gives a unit of $\mathrm{m\,s^{-2}}$, something has gone wrong: that is an acceleration unit, not a speed unit.

For Edexcel command words, this changes the kind of answer you give. “State” the SI unit if that is all the question asks. But if you are asked to “derive” or show a unit, the marks come from the route: start from the equation and combine the units step by step.

Prefixes and powers

A prefix multiplies a unit: $\mathrm{k}=10^3$, $\mathrm{M}=10^6$, $\mathrm{G}=10^9$, $\mathrm{c}=10^{-2}$, $\mathrm{m}=10^{-3}$, $\mu=10^{-6}$, $\mathrm{n}=10^{-9}$ and $\mathrm{p}=10^{-12}$. Thus $250\,\mathrm{mA}=0.250\,\mathrm A$ and $3.0\,\mu\mathrm s=3.0\times10^{-6}\,\mathrm s$. Case matters: m is milli, M is mega.

Convert a length before squaring or cubing it: $1\,\mathrm{mm^2}=(10^{-3}\,\mathrm m)^2=10^{-6}\,\mathrm{m^2}$, while $1\,\mathrm{cm^3}=10^{-6}\,\mathrm{m^3}$. Standard form uses one non-zero digit before the decimal point, such as $0.000024\,\mathrm m=2.4\times10^{-5}\,\mathrm m$. An order-of-magnitude estimate keeps the scale, about $10^{-5}\,\mathrm m$, without claiming measurement precision.

Estimate Before You Trust a Number

Physicists estimate before they calculate exactly. An estimate is not a guess with no discipline. It is a deliberately rough model using sensible sizes, rounded values and units. Its job is to catch impossible answers early.

Suppose a trolley rolls along a bench. Before timing anything, you can make a rough picture:

That is not meant to be the final answer. It is a scale. A final measured speed of $0.47\,\mathrm{m\,s^{-1}}$ would feel plausible. A final value of $47\,\mathrm{m\,s^{-1}}$ would be a warning sign, because that is faster than a car in a town and wildly too fast for a small classroom trolley.

Here is a worked estimate with a decision point.

Common unit error:

“The trolley travelled 120 cm in about 2 s, so the speed is $60\,\mathrm{m\,s^{-1}}$.”

The physical story says the trolley moved only a little more than a metre in a couple of seconds. That should be around walking pace, not motorway speed. The unit check shows the error:

The student divided $120$ by $2$ but forgot that the $120$ was in centimetres. The estimate caught the problem before the answer was accepted.

A good estimation route is:

1. Name the quantity you want.
2. Choose simple, plausible values in SI units.
3. Combine them with the same relationship you would use in the exact calculation.
4. Ask whether the result belongs in the real situation.

Here the important word is “use”: an estimate should help you solve or judge a problem. It is not a decorative line before the real work.

Choose the Method Before the Data

Practical physics starts before the first reading. You decide what to measure, which apparatus can measure it, and what limitations that apparatus will bring.

For the trolley speed problem, the measured quantities are:

A ruler marked every $1\,\mathrm{mm}$ has a resolution of $1\,\mathrm{mm}$. A stopwatch displaying $0.01\,\mathrm{s}$ has a resolution of $0.01\,\mathrm{s}$. Resolution matters, but it is not always the biggest limitation. If a human starts and stops a stopwatch, reaction time can produce a much larger spread than $0.01\,\mathrm{s}$.

That is why repeats matter. Repeated readings help you see random effects: small unpredictable variations that make readings scatter. Taking a mean can reduce their influence. But repeats do not cure a systematic error. If the distance marks are actually $2\,\mathrm{cm}$ too close together because the tape was misread, repeating the timing many times will not fix the wrong distance.

A practical method should therefore be described as a chain:

For the trolley:

• Measure a longer distance between two clear marks, because the same timing reaction error is then a smaller percentage of the journey time.
• Use repeated timings and calculate a mean, because individual stopwatch readings scatter.
• Release the trolley from the same point each time, because changing the starting point changes the speed being measured.
• If light gates are available, use them to reduce human reaction-time uncertainty, but still check that each gate is aligned with the trolley interrupt card.

This is how the method transfers to unfamiliar experiments. Even if the apparatus changes, the thinking route is stable: what quantity is needed, what instrument measures it, what limits the reading, and how will the final value be calculated?

For an Edexcel “Explain” question, do not just name an improvement. Link it to the effect. "Use a longer distance" is only half an answer. "Use a longer distance so the same reaction-time uncertainty is a smaller percentage of the measured time" is the physics.

Worked Example: Speed With Uncertainty

Now put the pieces together. A trolley travels between two marks. The distance between the marks is:

Repeated timings are:

The task is to determine the average speed with an uncertainty.

Decision point 1: convert to SI units.

Decision point 2: choose the time value. The readings scatter much more than the stopwatch display resolution, so use the mean and estimate uncertainty from the spread.

Record this as $2.25\,\mathrm{s}$, because the readings do not justify every calculator digit.

For the uncertainty in the repeated time, use half the range:

Decision point 3: calculate speed from the measured distance and measured time.

Decision point 4: combine uncertainties using percentages, because the calculation divides two measured quantities.

Convert that percentage into an absolute uncertainty in the final speed:

A sensible final result is:

Notice the representation shift. The raw data table became a mean and half-range. The equation gave the derived unit $\mathrm{m\,s^{-1}}$. The percentage uncertainties told us which measurement limited the result most: timing at $2.7\%$, not distance at $0.42\%$.

The common mistake is to write:

That looks impressive but is less honest. The extra digits are calculator precision, not measurement precision. A calculated answer should not pretend to know more than the measurements know.

In an Edexcel “Calculate” answer, show the substitution and include the unit. In a “Determine” answer, the route matters too: you may need to choose the mean, choose the uncertainty method, and then reach the quantitative result.

What the Uncertainty Lets You Claim

A measurement does not usually give a single perfect value. It gives a best value and a range of reasonable values.

For the trolley worked example:

That means the result is being treated as an interval:

If a calibrated light gate gives $0.54\,\mathrm{m\,s^{-1}}$ for the same motion, the student's result is consistent with it because $0.54\,\mathrm{m\,s^{-1}}$ lies inside the interval. If the calibrated value were $0.70\,\mathrm{m\,s^{-1}}$, the student's result would not be consistent, and you would look for a method problem.

This is where several everyday words need their physics meanings.

Precision and accuracy are not the same. Darts clustered tightly away from the bullseye are precise but not accurate. Widely scattered darts centred around the bullseye may be more accurate on average but less precise.

For measurements:

• Large scatter in repeated readings suggests random effects and poor precision.
• A zero error, misaligned scale, or consistently late start can cause a systematic error.
• Repeats can reveal random scatter, but they do not remove a systematic error.

Graph evidence follows the same idea. If a straight-line relationship is expected to pass through the origin but the best-fit line has a clear intercept, that can suggest a systematic error. If points are widely scattered about the best-fit line, that suggests larger random uncertainty. You do not need a graph for the trolley example, but the reasoning transfers.

For an Edexcel “Evaluate” question, use evidence and make a judgement. A good answer does not just say "accurate" or "not accurate"; it refers to the uncertainty interval and the comparison value.

Combining Measurement Uncertainties

The equation used to calculate a result decides how measurement uncertainties combine. Use these simple maximum-uncertainty estimates when the uncertainties are small compared with the readings. They describe a conservative range, not a formal statistical confidence interval.

For a sum or difference, add absolute uncertainties. A length measured between $x_1=12.0\pm0.1\,\mathrm{cm}$ and $x_2=38.0\pm0.1\,\mathrm{cm}$ is $26.0\pm0.2\,\mathrm{cm}$. Subtracting readings does not subtract their uncertainties: either endpoint could move the length in the opposite direction.

For a product or quotient, add fractional or percentage uncertainties, as in the speed calculation. For a power, multiply the percentage uncertainty by the magnitude of the power. If a circular wire has $d=0.40\pm0.01\,\mathrm{mm}$, its diameter uncertainty is 2.5%. Since $A=\pi d^2/4$, its area uncertainty is approximately 5%. The exact constants $\pi$ and 4 add no measurement uncertainty.

Keep extra digits during working; round the uncertainty sensibly and match the decimal place of the quoted value to it. An answer such as $0.50\pm0.02\,\mathrm{m\,s^{-1}}$ tells a reader what range is justified; six decimal places from a calculator do not demonstrate accuracy. When comparing two measured values, consider the uncertainty of both. Overlapping intervals support consistency under the stated model; they do not prove that systematic error is absent.