Distance-Time Graphs

The fundamental relationship is defined by the formula: $\text{Speed} = \frac{\Delta \text{Distance}}{\Delta \text{Time}}$.

Linearity: A straight line indicates that the object is moving at a constant speed, meaning it covers equal distances in equal intervals of time.

Steepness: The steeper the slope of the line, the higher the speed of the object. A perfectly vertical line is physically impossible as it would imply infinite speed.

Horizontal Lines: A slope of zero (a flat horizontal line) indicates that the distance is not changing over time, meaning the object is stationary.

Calculating Constant Speed: To find the speed from a straight-line segment, select two points $(t_1, d_1)$ and $(t_2, d_2)$ and calculate the gradient using $v = \frac{d_2 - d_1}{t_2 - t_1}$.

Interpreting Curves: If the graph is curved, the speed is changing. An upward curve (increasing gradient) represents acceleration, while a downward curve (decreasing gradient) represents deceleration.

Instantaneous Speed: To find the speed at a specific moment on a curve, draw a tangent to the curve at that point and calculate the gradient of that tangent line.

Average Speed: This is calculated by dividing the total distance traveled by the total time taken for the entire journey, regardless of stops or speed changes.

Check the Axes: Always verify the units on both axes before performing calculations. Converting minutes to seconds or kilometers to meters is a common requirement.

Identify Segments: Break complex journeys into distinct segments (e.g., constant speed, rest, returning) and analyze each part separately.

Tangent Precision: When drawing a tangent to find instantaneous speed, ensure the line just touches the curve at the specified point and extends far enough to allow for accurate coordinate reading.

Sanity Check: If a line is steeper than another, the calculated speed must be higher. If the line is horizontal, the speed must be zero. Always check if your numerical answer matches the visual trend.

Stationary vs. Constant Speed: A common error is assuming a horizontal line means constant speed. On a distance-time graph, a horizontal line means the distance is constant, so the object is not moving.

Negative Slopes: Students often think a negative slope means the object is slowing down. In reality, it means the object is moving in the opposite direction (returning to the start).

Curvature Interpretation: Confusing a curve on a distance-time graph with a curve on a velocity-time graph. On a distance-time graph, any curve indicates changing speed (acceleration or deceleration).