Plot, interpret and linearise graphs to extract physical quantities
AQA A-Level Physics 1
📏Plot graphs with correctly labelled axes and units
📐Draw lines of best fit and calculate gradient
➖Find the y-intercept and interpret its physical meaning
〰️Draw error bars and determine uncertainty in gradient
🔄Linearise non-linear relationships for graph plotting
🔍Identify anomalous results from a scatter of points
Plotting Graphs: The Basics
A well-drawn physics graph communicates your data clearly and allows accurate extraction of physical quantities. Every graph must follow these conventions:
Axes: The independent variable goes on the x-axis; the dependent variable on the y-axis. Both axes must be labelled with the quantity and its unit in the form: Quantity / unit (e.g. "Time / s", "Voltage / V").
Scale: Use a scale that makes the plotted points occupy at least half the available grid area. Scale must be linear and easy to read — avoid awkward scales like 1:3 or 1:7.
Plotting: Mark each point with a small cross (×) or dot with a circle. Plot all given data points. Check each one.
An anomalous result is a data point that does not fit the overall trend. It should be identified and, if caused by an experimental mistake, excluded from the line of best fit (but still plotted).
Lines of Best Fit & Gradient
A line of best fit (also called a trend line) is drawn so that the data points are balanced either side — approximately equal numbers of points above and below the line. It should pass through or near the majority of points; do not force it through the origin unless the physics demands it.
Gradient: Choose two well-separated points on the line (not data points) and calculate:
gradient m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)
Use as large a triangle as possible to minimise the effect of reading errors. Always include units: the gradient has units of (y-axis units) / (x-axis units).
y-intercept c: The value of y when x = 0. Read from the graph where the line crosses the y-axis, or calculate using y = mx + c rearranged to c = y − mx.
The equation of the line y = mx + c mirrors physical equations. For example, if you plot V against I, the gradient = R (Ohm's law: V = IR + 0, so c = 0).
Error Bars and Uncertainty in Gradient
Error bars show the uncertainty in each data point. Horizontal error bars represent uncertainty in x; vertical error bars represent uncertainty in y. The length of each bar = 2 × absolute uncertainty (one uncertainty either side of the point).
Line of best fit: Must pass through all error bars (or as many as possible).
Worst-case lines: Draw the steepest possible line and the shallowest possible line through the error bars. Calculate the gradient of each.
Uncertainty in gradient = (mmax − mmin) / 2
Similarly, the uncertainty in the y-intercept can be found from the spread of intercepts of the worst-case lines.
In AQA exams, you may be asked to draw worst-case lines and calculate the uncertainty in gradient — make sure your worst-case lines still pass through ALL error bars.
Linearising Non-Linear Relationships
Many physical relationships are not linear. By manipulating the equation algebraically, we can plot a straight-line graph. This makes it easy to verify the relationship and extract constants accurately.
Method: Identify the physical equation. Rearrange it into the form y = mx + c. Decide what to plot on each axis; gradient and intercept give the physical constants.
Relationship
Plot y
Plot x
Gradient
Intercept
T² = 4π²L/g
T²
L
4π²/g
0
v² = u² + 2as
v²
s
2a
u²
E = hf − φ
Ek
f
h
−φ
I = I₀e^(−x/λ)
ln I
x
−1/λ
ln I₀
For exponential relationships y = Ae^(kx), take natural logs of both sides: ln y = kx + ln A. Plot ln y vs x to get a straight line with gradient k and intercept ln A.
A student plots extension x (mm) against force F (N) and gets a straight line through the origin with points at (2.0, 40), (4.0, 80), (6.0, 120). Calculate the spring constant k.
1Hooke's law: F = kx → rearrange → k = F/x = gradient of F vs x graph
2Choose two points on the line: (2.0, 40) and (6.0, 120)
4Convert: 20 N mm⁻¹ = 20 × 10³ N m⁻¹ = 20 000 N m⁻¹
Spring constant k = 2.0 × 10⁴ N m⁻¹
In a pendulum experiment, T² values and lengths L are: (0.20 m, 0.81 s²), (0.40 m, 1.61 s²), (0.60 m, 2.42 s²), (0.80 m, 3.22 s²). A graph of T² vs L gives a straight line. Find g.
5. A graph of ln I vs x gives gradient −0.35 cm⁻¹. If I = I₀e^(−μx), what is μ?
1. A student investigates how the resistance R of a wire varies with its length L. They plot R (Ω) on the y-axis against L (m) on the x-axis. The graph is a straight line with gradient 4.8 Ω m⁻¹ and passes through the origin. Using ρ = RA/L and cross-sectional area A = 7.85 × 10⁻⁷ m², calculate the resistivity ρ.
R = (ρ/A) × L → gradient = ρ/A. ρ = gradient × A = 4.8 × 7.85 × 10⁻⁷ = 3.77 × 10⁻⁶ Ω m ≈ 3.8 × 10⁻⁶ Ω m. This is consistent with nichrome wire.
2. A student plots v² (y-axis, m² s⁻²) vs s (x-axis, m) to find acceleration. The worst-case gradients are 5.2 and 4.4 m s⁻². The best-fit gradient is 4.8 m s⁻². What is the acceleration and its absolute uncertainty?
v² = u² + 2as → gradient = 2a → a = gradient/2. Best: a = 4.8/2 = 2.4 m s⁻². Uncertainty in gradient = (5.2 − 4.4)/2 = 0.4 m s⁻². Uncertainty in a = 0.4/2 = 0.2 m s⁻². Answer: a = 2.4 ± 0.2 m s⁻².
3. Explain why the line of best fit should not always be forced through the origin, even when the physical theory predicts a directly proportional relationship.
A non-zero intercept may indicate a systematic error in the experiment (e.g. a zero error on a meter, or heat losses). Forcing the line through the origin would mask this error and produce an incorrect gradient. You should draw the best-fit line through the data points as they are, and then discuss the intercept in your analysis — a non-zero intercept is itself useful information about systematic errors in the experiment.