Physics: Systematic Error and Relative Uncertainty in Kinematics

Kinematics is the study of motion—how position, velocity, and acceleration change over time—and the velocity-time graph is its most powerful visual tool. For an underwater ROV moving along a pipeline, the graph splits into three distinct phases: constant acceleration from rest, steady cruising, and constant deceleration back to rest. Each phase has a simple geometric interpretation: the area under the graph gives displacement, while the slope gives acceleration. In this journey, the acceleration is found from the change in velocity divided by the time interval (a = Δv/Δt), and the total displacement is the sum of the areas of a triangle, a rectangle, and another triangle. But real measurements are never perfect. Here, the position sensor has a systematic error of +2.5 m—it always reads slightly too high. This fixed offset shifts the measured displacement and average velocity (v = s/t) upward, but crucially, its impact depends on the journey length. When the true displacement doubles, the same absolute error is spread over more distance, so the relative error (percentage error = absolute error / true value × 100) shrinks. This reveals a core idea: systematic errors affect accuracy, not precision, and their relative significance decreases as the measured quantity grows—a relationship that connects the graph’s geometry to the reliability of your final result.