Motion graphs (displacement–time & velocity–time)

What the gradient and the area mean on each kind of motion graph — gradient of displacement–time is velocity, gradient of velocity–time is acceleration, area under velocity–time is displacement — and how to model a journey as a graph and read unknowns off its area.

Mechanics · Kinematics

Motion graphs (displacement–time & velocity–time)

Why it works

A motion graph turns a whole journey into one picture. There are two kinds, and the single most important skill in this topic is keeping straight what the gradient and the area mean on each — because they mean different things.

Displacement–time graph (

s

up the side,

t

along the bottom). The graph tells you where the particle is at each instant. Its gradient is the velocity — because velocity is exactly the rate at which displacement changes,

v = \dfrac{ds}{dt}

. So:

a straight line = constant velocity (steeper = faster);
a horizontal line = gradient $0$ = the particle is at rest (still moving

through time, but not through space);

a negative gradient = moving back towards the start;
a curve = changing velocity, i.e. acceleration.

The area under a displacement–time graph means nothing — don't compute it.Velocity–time graph (

v

up the side,

t

along the bottom). Now two facts matter, and this is where the marks live:

The gradient is the acceleration, $a = \dfrac{dv}{dt}$ — a straight line is

constant acceleration, a horizontal line is constant velocity, a downward slope is deceleration.

The area between the line and the time-axis is the displacement. Area above

the axis is positive displacement; area below the axis is negative (the particle moved backwards). Add the signed areas and you get displacement; add their sizes and you get distance travelled — the same distinction as in calculus kinematics ([[kinematics.integrating-motion]]).When the acceleration is constant the velocity–time graph is made of straight lines, so the area is just triangles, rectangles and trapezia — no calculus needed. That is exactly why the suvat equations exist ([[kinematics.constant-acceleration]]):

s = \tfrac12(u+v)t

is the area of the trapezium under the line.

Modelling a journey. A typical journey — speed up, hold a steady speed, slow to a stop — is a trapezium like the one above. The power of this is that the total area equals the total distance, so if you're told the distance you can write the area in terms of the unknown times and solve for them. Set up the area, set it equal to the given distance, and the algebra hands you the answer.

Speed–time vs velocity–time — a real distinction. Speed is the size of the velocity, so it is never negative: a speed–time graph never dips below the axis, and the area under it is the distance travelled. A velocity–time graph can go negative (motion in the negative direction), and its signed area is displacement. Read the axis label before you decide which one you've got.