Distance-Time Graphs

• Gradient as Speed: The most critical principle is that the gradient (slope) of the line on a distance-time graph represents the speed of the object.

• Mathematical Derivation: Because speed is defined as distance divided by time ($v = s/t$), and the gradient is the change in the vertical axis divided by the change in the horizontal axis ($\Delta y / \Delta x$), the gradient directly yields the speed value.

• Slope Interpretation: A steeper slope indicates a higher speed, while a shallower slope indicates a lower speed. A slope of zero (horizontal line) indicates the object is not moving.

Calculating Speed from a Linear Graph

• Step 1: Select Points: Choose two distinct points on a straight section of the graph to ensure accuracy.
• Step 2: Determine Changes: Calculate the change in distance ($\Delta y$) and the change in time ($\Delta x$) between these two points.
• Step 3: Apply Formula: Use the gradient formula: $Speed = \frac{y_2 - y_1}{x_2 - x_1}$.

Calculating Instantaneous Speed from a Curve

• Step 1: Identify the Point: Locate the specific time on the curve where the speed needs to be measured.
• Step 2: Draw a Tangent: Draw a straight line that just touches the curve at that single point, ensuring the angles on both sides of the point are balanced.
• Step 3: Calculate Tangent Gradient: Find the gradient of this tangent line using the standard $\Delta y / \Delta x$ method to find the speed at that exact moment.

• Stationary vs. Constant Speed: It is vital to distinguish between a horizontal line and a diagonal line. A horizontal line means distance is not changing (speed is zero), whereas a diagonal line means distance is changing at a steady rate.

• Distance vs. Displacement: While distance-time graphs usually show total distance traveled (always positive or increasing), displacement-time graphs can show an object returning to its start (line moving back toward the x-axis).

• Check the Units: Always verify the units on both axes before calculating. If distance is in kilometers and time is in minutes, you may need to convert to meters and seconds to provide a standard $m/s$ answer.

• The Large Triangle Rule: When calculating gradients, draw a large triangle on the graph. Using a larger portion of the line reduces the percentage error in your measurements and is often a requirement for full marks in practical exams.

• Sanity Check: If a graph shows a very steep line, the calculated speed should be a high number. If your calculation results in a small number for a steep slope, re-check your division order.

• Tangent Accuracy: When drawing tangents for curved graphs, use a transparent ruler to ensure the line is perfectly balanced against the curve at the point of interest.