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A detective has only a dashcam's two graphs — one of position over time, one of velocity over time — and no video. From the shapes alone she states exactly how fast the car was going at the crash and how far it had travelled. How can two squiggly lines reveal both?
A detective has only a dashcam's two graphs — one of position over time, one of velocity over time — and no video. From the shapes alone she states exactly how fast the car was going at the crash and how far it had travelled. How can two squiggly lines reveal both?
A motion graph hides its answers in its geometry: the steepness (slope) of a line and the space (area) beneath it. Learn to read those two features and a graph tells you velocity, acceleration, and distance at a glance.
A motion graph stores its information in two geometric features: slope (steepness) and area (the region underneath). On a position–time (d–t) graph the slope equals the velocity. On a velocity–time (v–t) graph the slope equals the acceleration, and the area beneath the line equals the distance travelled.
Read graphs by shape. A flat d–t line = at rest (zero slope = zero velocity); a straight slanted d–t line = constant velocity (constant slope); a curve that gets steeper = speeding up. On the v–t graph, a flat line = constant velocity (zero slope = zero acceleration); a line sloping up = speeding up; sloping down = slowing down; and the shaded area underneath always gives the distance. For a straight-line region the area is just a rectangle or triangle: distance = ½ × base × height for a triangle, or base × height for a rectangle. The single most common slip is forgetting that a flat line means OPPOSITE things on the two graphs: on a d–t graph flat = stopped, but on a v–t graph flat = cruising at a steady speed. Always check which graph you are reading before you judge the motion. **Limiting case:** flattens the v–t graph and straightens x–t; uniform tilts v–t into a straight ramp — recognise these two shapes and most exam graphs decode instantly. **Connect it:** the three graphs are one motion differentiated twice — the slope of x–t is the v–t value, the slope of v–t is the a–t value, and going backwards, areas integrate you up the chain.
Reading a graph's height as the quantity itself — on a d–t graph the SLOPE is velocity, and on a v–t graph the slope is acceleration while the AREA is distance.