Constant acceleration (suvat)

The five suvat equations, why they all fall out of the velocity–time graph (gradient = acceleration, area = displacement), and the one condition that makes them valid — constant acceleration.

Mechanics · Kinematics

Constant acceleration (suvat)

The five suvat equations, why they all fall out of the velocity–time graph (gradient = acceleration, area = displacement), and the one condition that makes them valid — constant acceleration.

Why it works

When acceleration is constant, the motion is simple enough to capture in a handful of formulas — the suvat equations, named for the five quantities they connect:

s

(displacement),

u

(initial velocity),

v

(final velocity),

a

(acceleration),

t

(time).

The whole set comes from one picture: the velocity–time graph. Constant acceleration means velocity changes at a steady rate, so the v–t graph is a straight line. Two facts about that line give you everything:

Its gradient is the acceleration. Gradient $= \dfrac{\text{change in } v}{\text{time}} = \dfrac{v - u}{t}$ , and rearranging gives

v = u + at.

The area under it is the displacement. (This is the integration idea from

variable motion, but here the shape is just a trapezium.) Area of the trapezium with parallel sides

u

and

v

and width

t

s = \tfrac{1}{2}(u + v)t.

Everything else is these two combined. Substitute

v = u + at

into the trapezium (or split the area into a rectangle

ut

plus a triangle

\tfrac{1}{2}(at)t

) to get

s = ut + \tfrac{1}{2}at^2,

and eliminate

t

between the first two to get

v^2 = u^2 + 2as.

So the suvat equations aren't five things to memorise blindly — they're the geometry of a straight-line v–t graph.Above, the line's gradient is the acceleration and the shaded area is the displacement — read the motion straight off the graph.

Method: list

s, u, v, a, t

; write down the three you know and the one you want; choose the equation that leaves out the fifth.

The one condition — and the trap. All of this needed the v–t graph to be a straight line, i.e. constant acceleration. The moment acceleration varies with time, the graph curves, suvat is wrong, and you must use calculus instead (see [[kinematics.integrating-motion]]). Two more mark-savers: always pick a positive direction and stick to it (a deceleration is then a negative

a

), and for vertical motion under gravity take

a = \pm g \approx 9.8\ \text{m s}^{-2}

with the sign matching your chosen direction.