Drawing Travel Graphs

The gradient of a travel graph represents the rate of change of the y-axis variable with respect to time. For a displacement-time graph, the gradient is velocity ($v = \frac{ds}{dt}$); for a velocity-time graph, the gradient is acceleration ($a = \frac{dv}{dt}$).

The area under the curve of a velocity-time graph represents the total change in displacement. This is derived from the integral of velocity over time: $s = \int v \, dt$.

Vector vs. Scalar: Displacement and velocity are vectors, meaning they can be negative (indicating direction). Distance and speed are scalars and are always represented by the absolute magnitude of these values.

Step 1: Axis Setup

• Draw the horizontal axis for time ($t$) and the vertical axis for the given variable ($s$ or $v$). Ensure units are consistent across all given data points.

Step 2: Plotting Key Coordinates

• Identify critical moments in the journey, such as the start, changes in acceleration, or when the object comes to rest. Mark these as $(t, s)$ or $(t, v)$ coordinates on the axes.

Step 3: Connecting the Points

• Use straight lines to represent constant rates (constant velocity on an $s$-$t$ graph or constant acceleration on a $v$-$t$ graph). Use horizontal lines to represent zero rates (stationary on $s$-$t$ or constant velocity on $v$-$t$).

It is vital to distinguish between how motion is represented on each graph type to avoid calculation errors.

A curved line on a displacement-time graph indicates changing velocity (acceleration), whereas a curved line on a velocity-time graph indicates changing acceleration (non-uniform acceleration).

• Check the Y-Axis: Always verify if the graph is displacement-time or velocity-time before performing calculations. Finding the area of an $s$-$t$ graph is a common mistake that yields no useful value.

• Initial Conditions: Do not assume every journey starts at the origin $(0,0)$. Check the problem statement for an initial displacement or initial velocity.

• Total Distance vs. Displacement: In $v$-$t$ graphs, total distance is the sum of the absolute areas (all treated as positive). Total displacement is the area above the axis minus the area below the axis.

• Uniform Motion: Look for keywords like 'uniformly' or 'at a constant rate' to justify drawing straight lines between points.

• Negative Gradients: A negative gradient on an $s$-$t$ graph means the object is moving back toward the origin, not necessarily slowing down. On a $v$-$t$ graph, a negative gradient always means deceleration (or acceleration in the negative direction).

• Instantaneous Rest: On a $v$-$t$ graph, the object is only stationary when the line touches or crosses the x-axis ($v=0$). A horizontal line at $v=5$ means the object is moving at a steady speed, not stopped.

• Units Mismatch: Ensure time is in seconds if velocity is in $m s^{-1}$. If time is given in minutes, convert it before plotting or calculating areas.