Kinematics and the SUVAT Equations

A-Level Maths · Mechanics

Modelling motion

Mechanics uses models with simplifying assumptions: a particle (mass at a point, ignore size), smooth (no friction), light (massless string/rod), inextensible string, and gravity g ≈ 9.8 m/s². Quantities are vectors (displacement, velocity, acceleration) or scalars (distance, speed).

The SUVAT equations

For motion in a straight line with constant acceleration:

v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
s = vt − ½at²

where s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time. List what you know and pick the equation with only one unknown.

Motion graphs

Displacement–time: gradient = velocity.
Velocity–time: gradient = acceleration; area under = displacement.

Use graph areas (triangles/trapezia) for problems with changing but piecewise-constant acceleration.

Vertical motion under gravity

An object in free fall has acceleration g downward. Choose a positive direction and be consistent with signs (e.g. up positive → a = −g). At the highest point, the velocity is momentarily zero.

Calculus for variable acceleration

When acceleration is not constant, use calculus:

v = ds/dt (differentiate displacement) and a = dv/dt.
s = ∫v dt and v = ∫a dt (integrate, using initial conditions to find the constant).

Worked example

A ball is thrown up at 14.7 m/s. How high does it rise? (g = 9.8, take up as positive so a = −9.8.)

At the top v = 0. Use v² = u² + 2as: 0 = 14.7² + 2(−9.8)s → 0 = 216.09 − 19.6s → s = 11.0 m. ✓

Common mistakes

Using SUVAT when acceleration is not constant (use calculus instead).
Sign errors in vertical motion — decide the positive direction first.
Forgetting velocity is zero at the highest point of vertical motion.

Exam tips

List s, u, v, a, t and choose the right SUVAT equation.
For variable acceleration, differentiate to go from s → v → a, integrate to go back.
Use velocity–time graph areas for displacement.

Key facts to remember

SUVAT equations apply only for constant acceleration; free fall uses a = g ≈ 9.8 m/s².
Velocity–time graph: gradient = acceleration, area = displacement.
Variable acceleration: v = ds/dt, a = dv/dt; s = ∫v dt, v = ∫a dt.