Distance-Time Graphs

Calculating Speed from a Gradient

• To calculate the speed from a distance-time graph, select two points on a straight-line section and determine the 'rise' (change in distance) and 'run' (change in time).
• Use the gradient formula: $speed = \frac{\Delta y}{\Delta x} = \frac{d_2 - d_1}{t_2 - t_1}$. Always ensure that the units used on the axes (e.g., km and minutes) are converted to standard units (m and s) if the answer is required in m/s.

Analyzing Complex Journeys

• Break the graph into distinct segments based on the shape of the line. Identify sections of constant speed (straight slopes), stationary periods (flat lines), and varying speed (curves).
• To find the total distance traveled, look at the final vertical value on the y-axis if the object only moves away from the origin. If the journey involves returning toward the origin, calculate the distance for each segment and sum them.

• Steepness vs. Speed: A common point of confusion is comparing two lines. A line that is closer to the vertical axis has a 'larger' gradient and thus represents a higher speed compared to a line closer to the horizontal axis.

• Distance vs. Displacement: While distance-time graphs only show how far an object has moved, displacement-time graphs can account for direction. A downward slope on a displacement graph indicates moving back toward the origin, whereas on a pure distance graph, distance is typically cumulative.

• The 'Large Triangle' Rule: When calculating gradients in exams, always draw a large gradient triangle that covers at least half of the line section. This reduces errors in reading coordinates and is often a specific requirement in mark schemes to award 'working out' marks.

• Unit Awareness: Always check the axis labels immediately. Exams often use non-standard units like kilometers and minutes to test your ability to convert values (e.g., $1\text{ km} = 1000\text{ m}$ and $1\text{ min} = 60\text{ s}$) before performing calculations.

• Reading the Story: Practice describing the motion in words based on the graph. For example, 'The object moves at a constant speed for 5 seconds, stops for 3 seconds, then moves faster at a constant speed' is a typical descriptive task.

• Sanity Check: After calculating a speed, ask yourself if it is realistic. A walking speed should be around $1-2\text{ m/s}$, while a car might be $20-30\text{ m/s}$. If your calculation results in $500\text{ m/s}$ for a student walking, re-check your unit conversions.

• Confusing Distance with Speed: Students often look at the 'height' of the line to determine speed. The height only tells you the distance; you must look at the 'steepness' (gradient) to determine the speed.

• Misinterpreting Curves: An upward curve is often mistaken for a generic 'increase'. Be specific: the gradient is increasing, therefore the speed is increasing (acceleration).

• Stationary vs. Constant Speed: A flat line does NOT mean constant speed; it means zero speed. Constant speed is represented by a straight diagonal line. This is a very common error in multiple-choice questions.