Topics

What the investigation has to produce

What the report has to contain

The report is built in three layers, and each one is added on top of the last. Knowing all three before you start is worth more than any amount of extra data.

Layer	What it means
The floor	Data collected over a reasonable range and number of values, a graph of the relationship, and a conclusion naming the type of relationship
The detail	The variables that significantly affect the result controlled, techniques that increase accuracy used and written up, and the actual mathematical relationship obtained from your data stated
The evaluation	A discussion of the critical issues — the equivalent of two good discussion points

The first row on its own is correct and unfinished: it says what shape the relationship has without saying what the relationship is, and without asking whether the data can be trusted.

The three things that make this standard different

The relationship must be non-linear — a straight-line result gives you nothing to linearise. Choose an investigation where the physics predicts a curve.
You must use at least five different values of the independent variable. Five is the minimum, not the target; more values make the shape of the relationship clearer.
The conclusion must name the relationship in terms of your own variables, not in generic letters.
- Weak: "y is proportional to x squared."
- Strong: "The period $T$ is proportional to the square root of the length $L$ , with $T = 2.0\sqrt{L}$ ."

Choosing an investigation that works

A good investigation of this kind has all of these:

A clearly non-linear relationship predicted by physics, so linearising has a point.
An independent variable you can change easily over a wide range with the equipment available.
A dependent variable you can measure precisely enough that the change between values is much bigger than the measurement uncertainty.
A small number of controllable variables, so a fair test is actually achievable.

Common choices that meet all four:

Investigation	Independent	Dependent	Expected relationship
Simple pendulum	length $L$	period $T$	$T \propto \sqrt{L}$
Ball rolling off a bench	height $h$	speed $v$	$v \propto \sqrt{h}$
Light from a lamp	distance $d$	illuminance / intensity	$\propto \dfrac{1}{d^2}$
Sound level from a source	distance $d$	intensity	$\propto \dfrac{1}{d^2}$
Stretched wire	length $l$	frequency of vibration	$\propto \dfrac{1}{l}$
Trolley down a ramp	height of ramp	speed at the bottom	$\propto \sqrt{h}$

Writing the aim

The aim names both variables and says you are looking for the mathematical relationship between them.
- Weak: "To investigate a pendulum."
- Strong: "To find the mathematical relationship between the length of a pendulum and its period of oscillation."
Do not state the answer in the aim. You are finding the relationship from the data, not confirming a memorised one.

Tips and Tricks

The single biggest cause of a wasted investigation is a linear result. If your graph of raw data comes out as a straight line, there is nothing to linearise and nothing to find, no matter how good the write-up is — check with your teacher before you collect the full data set that the physics predicts a curve.

Know the shape of the report before you start writing, not after. It is built in three layers: carrying out the investigation and processing the data, then reasons and justifications at every stage, then evaluation of the critical issues affecting the reliability and validity of your findings. Knowing which sentences do which job lets you write them deliberately.

Do a quick pilot run before committing to a full data set — three or four rough readings across the widest range you can manage. It takes ten minutes and it reveals a linear relationship, an unworkable range, or an unusable instrument while there is still time to change the plan.

Worked ExampleTurning a topic into a workable investigation

A student wants to investigate "how a ball bounces". Turn this into a workable investigation, and explain each decision.

Step 1 — Check it can give a non-linear relationship

"How a ball bounces" is a topic, not an investigation. First, find a pair of variables within it whose relationship the physics predicts to be non-linear.

Dropping a ball from height $h$ and measuring the speed at which it hits the ground gives $v = \sqrt{2gh}$ , so $v \propto \sqrt{h}$ — a square-root relationship, which is non-linear. Suitable.

By contrast, dropping from height $h$ and measuring the rebound height gives a relationship that is very close to directly proportional for a given ball. That would produce a straight line and there would be nothing left to find, however carefully it was done.

Step 2 — Write the aim precisely

Both variables are named, and the aim asks for the relationship rather than stating it.

Step 3 — Identify the variables

Independent: drop height $h$ , measured with a metre rule fixed vertically.
Dependent: impact speed $v$ , measured with a light gate placed just above the floor.
Controlled: the same ball throughout (mass and diameter affect air resistance); released from rest each time rather than thrown; the same floor surface and the same light-gate position.

Step 4 — Plan the values

Eight heights from $0.20$ m to $1.60$ m in $0.20$ m steps — six or more values across a wide range, so the curvature is unmistakable. Three repeats at each height, averaged.

Step 5 — Plan the analysis before collecting data

Plot $v$ against $h$ first, expecting a curve that flattens off. Then linearise by plotting $v$ against $\sqrt{h}$ , or $v^2$ against $h$ — either should give a straight line through the origin, and the gradient of the $v^2$ against $h$ plot should equal $2g \approx 19.6$ m s⁻².

$\text{A non-linear relationship, wide range, controlled variables, and a planned analysis}$