Prefixes and Graph Skills

Ever wondered why we use "kilo" in kilogram or "mega" in megabyte? Unit prefixes are shortcuts that make huge or tiny numbers manageable. From tera (10¹²) down to pico (10⁻¹²), these prefixes help physicists communicate measurements clearly.

When creating graphs, you're basically telling a visual story with your data. The independent variable (what you control) goes on the x-axis, whilst the dependent variable (what you measure) sits on the y-axis. Your graph should fill at least 75% of the available space—nobody wants to squint at a tiny plot in the corner!

Top tip: When calculating gradients, draw your triangle across at least 75% of the straight line. This reduces the impact of small measurement errors and gives you a more reliable result.

Remember to label your axes properly with both the quantity and unit $like "Velocity/ms⁻¹"$, and always use a line of best fit rather than connecting every single point.

Understanding Measurements and Uncertainties

Every measurement you take in physics has some level of uncertainty—and that's perfectly normal! The key difference lies between random errors (which vary unpredictably) and systematic errors (which consistently shift your results in one direction).

Random errors pop up because of environmental factors or difficulty reading instruments. You can't eliminate them completely, but taking multiple measurements and calculating the mean helps reduce their impact. Systematic errors, however, come from faulty equipment or poor technique—these you can actually fix.

Accuracy tells you how close you are to the true value, whilst precision shows how consistent your repeated measurements are. Think of it like darts: accurate throws cluster around the bullseye, precise throws cluster together (but might miss the target entirely).

Remember: After discarding outliers, your absolute uncertainty is ± (range of measurements ÷ 2). If you've only got one measurement, use the resolution of your measuring equipment instead.

Percentage uncertainty helps you compare the reliability of different measurements: just divide absolute uncertainty by the actual value, then multiply by 100.

Combining Uncertainties and Graphical Analysis

When you're combining measurements in calculations, uncertainties don't just disappear—they follow specific rules. Add absolute uncertainties when you're adding or subtracting quantities. Add percentage uncertainties when multiplying or dividing. If you're raising something to a power, multiply the percentage uncertainty by that power.

Dealing with error bars on graphs might seem fiddly, but they're crucial for honest data analysis. Your line of best fit must pass within all error bars, with roughly equal numbers of data points on either side. Ignore any anomalous data—those weird results that clearly don't fit the pattern.

To find uncertainty in your gradient, draw the line of worst fit—either the shallowest or steepest line that still passes through all error bars. The difference between your best and worst gradients gives you the absolute uncertainty.

Pro tip: The line of worst fit is your reality check. It shows the range of possible gradients your data could reasonably support, making your conclusions more robust.

Scalars, Vectors, and Vector Addition

Not all quantities are created equal in physics. Scalar quantities have magnitude but no direction—like temperature or speed. Vector quantities have both magnitude and direction—like velocity or force. This distinction matters because vectors follow special rules for addition and subtraction.

Adding parallel vectors is straightforward: same direction means simple addition, opposite directions mean subtraction. But when vectors meet at angles, you need a vector triangle. Start by drawing the first vector to scale, then draw the second vector with its tail touching the first vector's head.

The resultant vector runs from the tail of your first vector to the head of your second vector. For precise calculations, use the cosine rule: a² = b² + c² - 2bc cos A. This mathematical approach gives you exact values rather than relying on careful drawing and measuring.

Key insight: Vector addition isn't just mathematical—it represents real physical situations, like combining forces or calculating displacement when you change direction during a journey.

Whether you're analysing forces on a bridge or calculating the path of a projectile, understanding vector addition gives you the tools to solve complex real-world problems.