Displacement & Velocity-Time Graphs

• Gradient as Velocity: The slope or gradient of a line on an $s-t$ graph represents the velocity of the object. A steeper gradient indicates a higher velocity, while a horizontal line indicates the object is stationary ($v = 0$).

• Linear vs. Non-linear Motion: A straight diagonal line represents constant velocity, meaning the object covers equal displacements in equal time intervals. A curved line indicates acceleration or deceleration, as the velocity (slope) is changing over time.

• Directional Interpretation: A positive gradient shows motion in the positive direction, while a negative gradient indicates the object is returning toward the origin or moving in the opposite direction.

• Gradient as Acceleration: The slope of a $v-t$ graph represents the acceleration ($a$) of the object. A constant upward slope indicates uniform acceleration, while a horizontal line indicates motion at a constant velocity ($a = 0$).

• Area as Displacement: The total area bounded by the graph line and the time axis represents the change in displacement. This is based on the relationship $s = \int v \, dt$, where the accumulation of velocity over time results in a change in position.

• Vector Considerations: Areas above the time axis represent positive displacement, while areas below the axis represent negative displacement. To find the total distance traveled, the absolute values of these areas are summed; to find total displacement, the signed values are combined.

Comparison of Graph Interpretations

• Distance vs. Displacement: On a $v-t$ graph, distance is the total area regardless of sign, whereas displacement is the vector sum (positive areas minus negative areas).

• Average vs. Instantaneous: The gradient of a chord between two points gives average velocity/acceleration, while the gradient of a tangent at a single point gives instantaneous values.

• Check the Axes First: Always verify if the vertical axis is displacement, velocity, or acceleration before interpreting the slope or area, as the same shape means different things on different graphs.

• Geometric Decomposition: When calculating the area under a $v-t$ graph, break complex shapes into simple rectangles, triangles, and trapeziums to minimize calculation errors.

• Sign Consistency: Pay close attention to motion below the x-axis. A negative velocity on a $v-t$ graph means the object is moving backward; ensure you subtract this area if the question asks for displacement.

• Units Verification: Ensure the units of the gradient (e.g., $m/s^2$) and area (e.g., $m$) match the physical quantities they represent. If time is in minutes, convert to seconds if the velocity is in $m/s$.

• Confusing $s-t$ and $v-t$ Slopes: Students often see a horizontal line on a $v-t$ graph and assume the object is stationary, when it is actually moving at a constant, non-zero velocity.

• Ignoring the Origin: Assuming every graph starts at $(0,0)$ can lead to errors in displacement. Always check the initial value on the y-axis.

• Curvature Misinterpretation: On an $s-t$ graph, a curve that gets steeper represents acceleration, while one that flattens out represents deceleration. Students often mix these up visually.