Distance-Time Graphs

• The mathematical definition of speed is the rate of change of distance with respect to time. On a graph, the rate of change between two quantities is represented by the gradient of the line connecting them.

• For any straight-line segment on a distance-time graph, the speed can be calculated using the formula for gradient: $speed = \frac{\text{change in distance}}{\text{change in time}} = \frac{\text{rise}}{\text{run}}$. This means that if the distance changes by $\Delta d$ over a time interval $\Delta t$, the speed is $v = \frac{\Delta d}{\Delta t}$.

• A steeper line indicates a larger change in distance over the same change in time, thus representing a higher speed. Conversely, a flatter line signifies a smaller change in distance over the same time, indicating a lower speed.

• A horizontal line on a distance-time graph signifies that the distance from the starting point is not changing over time. This implies that the object is stationary or at rest, as its speed is zero (gradient = 0).

• Moving away from the start: A line segment with a positive gradient indicates that the object's distance from the starting point is increasing. The object is moving away from its origin.

• Moving towards the start: A line segment with a negative gradient indicates that the object's distance from the starting point is decreasing. This means the object is returning towards its origin.

• Calculating instantaneous speed: For any straight-line segment, select two points $(t_1, d_1)$ and $(t_2, d_2)$ on the line. The speed for that segment is then calculated as $speed = \frac{d_2 - d_1}{t_2 - t_1}$. Ensure consistent units for distance and time.

• Calculating overall average speed: For a journey composed of multiple segments, including periods of rest, the overall average speed is determined by dividing the total distance traveled by the total time taken for the entire journey. This includes any time spent stationary.

Formula: $Overall\ Average\ Speed = \frac{Total\ Distance\ Travelled}{Total\ Time}$

• Misinterpreting horizontal lines: A common error is to think a horizontal line on a distance-time graph means constant speed. It actually means the object is stationary (speed is zero), as its distance from the origin is not changing.

• Excluding rest time from average speed: When calculating overall average speed, students sometimes forget to include the time intervals during which the object was at rest. The formula requires the total time for the entire journey, including stops.

• Confusing distance with displacement: While distance-time graphs typically show cumulative distance, a negative gradient implies movement in the opposite direction. If the question implies displacement, a negative gradient means moving back towards the origin, reducing the net displacement.

• Incorrectly calculating gradient: Errors can occur if the 'rise' and 'run' are not correctly identified or if the scale of the axes is misread. Always ensure that the units are consistent and that the change in distance and time are calculated accurately.

• Always check the axes: Before attempting any calculations or interpretations, confirm that the vertical axis represents distance and the horizontal axis represents time. Also, note the units used for each axis.

• Break down complex journeys: For journeys with multiple segments, analyze each segment individually to determine the motion (moving away, resting, moving towards, speed). This helps in systematically answering multi-part questions.

• Double-check calculations: Questions involving distance-time graphs often have interdependent parts. A mistake in an early calculation (e.g., speed of one segment) can lead to incorrect answers for subsequent parts (e.g., overall average speed).

• Verify reasonableness of answers: After calculating a speed or time, consider if the answer makes sense in the context of the graph. For example, a very high speed for a short, flat segment would indicate an error.