Displacement-Time Graphs

What the gradient means — the gradient of a displacement–time graph is the velocity, because velocity is rate of change of displacement.

Straight line — constant velocity. A steeper line means a greater speed, and a horizontal line means the object is stationary.

Negative gradient — motion in the negative direction, back towards the origin. The displacement is decreasing, which is not the same as decelerating.

Curved line — changing velocity, which means the object is accelerating. A curve getting steeper shows speeding up; a curve flattening shows slowing down.

Instantaneous velocity from a curve — draw a tangent to the curve at the point of interest and find the gradient of that tangent, using as large a triangle as the tangent allows.

Example: drawing the tangent is the part that costs marks. It must touch the curve at exactly the point of interest and match its direction there, and the triangle used to find its gradient should span most of the tangent's length, not a few millimetres of it.

Average versus instantaneous velocity — average velocity over an interval is the gradient of the straight chord joining the two end points. Instantaneous velocity is the gradient of the tangent at a single instant. On a curve these differ, and a question asking for "the velocity at t = 3.0 s" means the tangent.

Example: a car accelerating from rest covers 100 m in 10 s. Its average velocity is 10 m s⁻¹, but its instantaneous velocity at t = 10 s is 20 m s⁻¹ — it was below average for the whole first half of the journey.

Displacement, not distance — a displacement–time graph can come back down to zero, meaning the object has returned to its starting point. The distance travelled keeps increasing throughout; only the displacement falls.