Position–Time Graphs

Objective: Interpret a straight line on a position–time graph.

For motion along one coordinate axis, a straight position–time graph has constant slope Δx/Δt and therefore constant velocity over its represented interval. A line rising with time has positive velocity, a falling line negative velocity, and a horizontal line zero velocity. A steeper line in absolute value represents a greater speed in that one-dimensional interval. The vertical intercept is the modeled initial position at t = 0, which can be positive, negative, or zero. A straight line does not reveal an object’s size or the path taken in dimensions not shown. Read axis scales and units before computing slopes.

Objective: Compute signed displacement and path distance from graph segments.

A piecewise position–time graph can describe intervals of motion with different velocities. A horizontal segment shows rest in the stated frame; a change from positive to negative slope marks reversal of direction, provided the object follows the represented one-dimensional path. Total displacement is final plotted position minus initial plotted position. Distance traveled across monotonic segments is the sum of absolute position changes on each segment. For x = 0 → 6 → 6 → 2 m, displacement is +2 m and distance is 6 + 0 + 4 = 10 m. A sharp corner is an idealization of a rapid velocity change, not a claim of physically infinite acceleration.

Objective: Read local slopes without treating graph height as speed.

A curved position–time graph has a slope that changes with time, so the one-dimensional velocity is not constant. A tangent line estimates instantaneous velocity at a selected point; the slope of a secant line between two points gives average velocity for that interval. Increasing graph height alone means position is increasing, not necessarily that speed is increasing. If a rising curve flattens, its positive velocity decreases; if it steepens, positive velocity increases. Data resolution and curve fitting limit precise tangent estimates. Acceleration can be inferred from how the velocity-related slope changes, but a graph of position is not directly a graph of acceleration.