The Gradient Principle states that the rate of change of the y-variable with respect to the x-variable (time) yields a new kinematic quantity. On a displacement-time graph, the gradient equals velocity, while on a velocity-time graph, the gradient equals acceleration.
The Area Principle applies specifically to velocity-time graphs, where the definite integral (or geometric area) between the graph and the time axis represents the change in displacement. This is derived from the relationship .
Linearity and Motion: A straight line on a travel graph indicates a constant rate of change. For displacement-time graphs, this means constant velocity; for velocity-time graphs, this means constant (uniform) acceleration.
Step 1: Axis Setup: Draw the horizontal axis for time and the vertical axis for the relevant kinematic variable. Ensure you include clear labels and the correct units (e.g., , , ).
Step 2: Plotting Key Coordinates: Identify specific moments in the journey, such as the start time, the end of a constant acceleration phase, or the moment an object stops. Mark these as or points on the grid.
Step 3: Connecting Segments: Use straight lines to connect points if the motion is described as 'uniform' or 'constant'. Use curves if the rate of change is variable (e.g., non-constant acceleration on a displacement-time graph).
Step 4: Working Backwards: If a final position or total distance is given but a time or velocity is missing, use the area or gradient formulas to solve for the unknown variable before completing the sketch.
| Feature | Displacement-Time Graph | Velocity-Time Graph |
|---|---|---|
| Gradient | Represents Velocity () | Represents Acceleration () |
| Horizontal Line | Object is Stationary () | Constant Velocity () |
| Area under Graph | No standard physical meaning | Represents Displacement () |
| Below x-axis | Object is behind the origin | Object is moving backwards |
Distance vs. Displacement: When drawing, remember that displacement can be negative (indicating direction), whereas a distance-time graph would only ever show positive values and non-decreasing total distance.
Speed vs. Velocity: On a velocity-time graph, a negative y-value indicates motion in the opposite direction. To find total distance from a velocity-time graph, treat all areas (above and below the axis) as positive and sum them.
Check the 'Uniform' Keyword: If a question mentions 'uniformly', it is a signal to draw a straight line on the velocity-time graph, as this implies constant acceleration.
Verify the Origin: Always check if the journey starts at the origin . Some scenarios describe an object starting at a specific displacement or initial velocity, which must be reflected on the y-axis.
Sanity Check Areas: After drawing a velocity-time graph, calculate the area of the shapes (triangles, rectangles, trapeziums) to ensure the total displacement matches the information provided in the text.
Labeling Intercepts: Ensure that points where the graph crosses the x-axis are clearly labeled, as these represent moments when the object is at the origin (s-t) or instantaneously at rest (v-t).
Confusing Graph Types: A common error is calculating the area under a displacement-time graph. Remember that area only has physical significance (displacement) on a velocity-time graph.
Ignoring Direction: Students often forget that a negative gradient on a displacement-time graph means the object is returning toward the origin, not necessarily slowing down.
Incorrect Shape for Acceleration: On a displacement-time graph, constant acceleration is represented by a curve (parabola), not a straight line. A straight line on an s-t graph indicates zero acceleration.
