Graph Skills and Data Analysis
Graph Skills and Data Analysis
Graphical analysis is a core skill in A-Level Physics. Graphs are used to identify relationships, determine physical constants, and test theoretical predictions. Proper graph technique is assessed in written papers and practical endorsement.
Plotting Graphs
Essential requirements:
- Title: Describe what is being plotted (e.g., "Voltage against current for a filament lamp")
- Axes: Label each axis with the quantity and unit (e.g., V / V or Voltage (V))
- Scales: Choose scales so the data occupies at least half the available graph area in each direction. Use simple scales (1, 2, 5, or multiples of 10 per division). Do not use awkward scales like 3 or 7 per division.
- Origin: Include the origin (0, 0) only if it is a valid data point or if the relationship predicts zero. Otherwise, start axes at appropriate values.
- Points: Plot points as small crosses (×) or dots with circles, not large blobs
- Error bars: Draw when uncertainty values are available
Lines of Best Fit
- Draw a smooth line or curve that best represents the trend
- The line should pass through or near the majority of points
- There should be roughly equal numbers of points above and below the line
- Do not connect the dots — this is not a best fit
- Anomalous points (outliers far from the trend) should be identified and circled but not used when drawing the line
Straight Line Graphs: y = mx + c
The most powerful technique in physics graphing is to linearise the relationship so that a straight line graph can be plotted. The gradient and intercept then give physical constants.
| Relationship | What to plot | Gradient | Intercept |
|---|---|---|---|
| v = u + at | v vs t | a | u |
| V = ε − Ir | V vs I | −r | ε |
| T² = (4π²/g)L | T² vs L | 4π²/g | 0 |
| ln A = ln A₀ − λt | ln A vs t | −λ | ln A₀ |
| log P = n log V + log k | log P vs log V | n | log k |
Determining the Gradient
1. Draw a large triangle on the line (not through data points — on the actual line)
2. Use points on the line far apart to minimise percentage uncertainty
3. Calculate: gradient = Δy/Δx = (y₂ − y₁)/(x₂ − x₁)
4. Include the units of the gradient (y-unit / x-unit)
5. For the uncertainty in the gradient, draw the steepest and shallowest acceptable lines through the error bars and find their gradients
Determining the Y-Intercept
- Read the y-intercept directly from the graph if the x-axis includes zero
- If the x-axis does not start at zero, calculate using: c = y₁ − m × x₁ (where (x₁, y₁) is a point on the line and m is the gradient)
Logarithmic Plots
Used to determine the power in a power law relationship or to linearise exponential relationships.
Power law: y = kxⁿ
Take logs: log y = n log x + log k
Plot log y vs log x: straight line with gradient n and intercept log k.
Exponential: y = Ae^(kx)
Take natural logs: ln y = kx + ln A
Plot ln y vs x: straight line with gradient k and intercept ln A.
Worked Example — Radioactive Decay
Activity measurements: A = 850, 620, 455, 332, 243 Bq at t = 0, 50, 100, 150, 200 s
Calculate ln A: 6.745, 6.430, 6.121, 5.806, 5.493
Plot ln A against t. The graph is a straight line confirming exponential decay.
Gradient = (5.493 − 6.745)/(200 − 0) = −1.252/200 = −6.26 × 10⁻³ s⁻¹
Therefore λ = 6.26 × 10⁻³ s⁻¹
t₁/₂ = ln 2 / λ = 0.693 / 6.26 × 10⁻³ = 110.7 s
y-intercept = 6.745 → A₀ = e^6.745 = 849 Bq (consistent with the initial measurement)
Testing Proportionality
Directly proportional (y ∝ x): Plot y vs x; must give a straight line through the origin.
Inversely proportional (y ∝ 1/x): Plot y vs 1/x; must give a straight line through the origin. Alternatively, plot y vs x and show it gives a rectangular hyperbola.
Important: A straight line that does not pass through the origin shows a linear relationship (y = mx + c) but NOT direct proportionality.
Using Graphs to Determine Quantities
| Graph feature | Quantity determined |
|---|---|
| Gradient of displacement-time | Velocity |
| Gradient of velocity-time | Acceleration |
| Area under velocity-time | Displacement |
| Area under force-displacement | Work done |
| Area under force-time | Impulse |
| Area under V-Q graph | Energy stored (capacitor) |
| Gradient of V-I graph | Resistance |
| Gradient of charge-time (capacitor) | Current |
Tangents to Curves
When the relationship is non-linear, the instantaneous rate of change at a point is found by drawing a tangent to the curve at that point and calculating the gradient of the tangent.
Method:
1. Mark the point of interest on the curve
2. Place a ruler so it touches the curve at that point without crossing it
3. Draw the tangent line (extend it well beyond the point)
4. Calculate the gradient of the tangent using a large triangle
Example: The gradient of a curved displacement-time graph at a specific time gives the instantaneous velocity at that moment.
Interpolation and Extrapolation
Interpolation: Reading values within the range of the data — generally reliable.
Extrapolation: Extending the line beyond the data range to predict values — less reliable as the relationship may change.
Evaluating Experimental Results
When comparing an experimental value to a known value:
Percentage difference = |experimental − accepted| / accepted × 100%
If this is within the experimental uncertainty, the result is consistent with the accepted value. If outside, look for systematic errors.