Translating Between Motion Graphs

Position, velocity, and acceleration graphs of the same trip look different because they're answering different questions. Position asks where. Velocity asks how fast and which way. Acceleration asks whether the velocity is changing. They're connected by a strict rule: the slope of the position graph at any instant equals the velocity at that instant, and the slope of the velocity graph equals the acceleration. Going the other direction, the area under the velocity graph gives the change in position, and the area under the acceleration graph gives the change in velocity.

To build a position graph from a velocity graph, track area, not height. If velocity is a flat positive line, equal time chunks add equal position chunks, so position climbs in a straight line with constant slope. If velocity is a sloped line, the area accumulated keeps growing differently as velocity grows, so position curves. A flat acceleration graph means velocity gains the same amount every second, which draws a straight slanted velocity line, and that slanted velocity line sweeps out growing area, which curves the position graph into a parabola.

This topic sits right between the single-variable definitions of velocity and acceleration and the next step, projectile motion, where you apply all three graphs to vertical and horizontal motion at once. A ball tossed straight up has a position graph shaped like an upside-down parabola, a velocity graph that is a straight line crossing zero at the peak, and an acceleration graph that stays flat at -9.8 m/s² the entire time, even at the peak.