Understand, use and derive the formulae for constant acceleration for…

The idea

For constant acceleration a along a straight line, velocity changes linearly with time: v = u + at. Displacement accumulates as s = ut + ½at², and eliminating t gives v² = u² + 2as; the average-velocity form s = ½(u + v)t follows from the trapezium area under a velocity–time graph. Derivation uses a = dv/dt and v = ds/dt with a constant, integrating from t = 0 with v = u and s = 0. In two dimensions each formula applies independently to the x- and y-components, so r = r₀ + u t + ½a t² and v = u + a t become vector equations; a projectile has a = −g ĵ while the horizontal component of velocity stays constant. Choose the equation omitting the unwanted quantity, fix a positive direction, and keep signs consistent.

What earns the marks

• States the five constant-acceleration formulae and identifies which variable each omits, so the correct equation can be selected for a given problem.
• Derives v = u + at by integrating a = dv/dt with constant a and initial condition v = u at t = 0.
• Derives s = ut + ½at² by integrating v = u + at with s = 0 at t = 0, and obtains v² = u² + 2as by eliminating t.
• Applies the formulae componentwise in two dimensions, writing r = r₀ + u t + ½a t² and v = u + a t as vector equations.
• Interprets gradient and area of velocity–time and displacement–time graphs as acceleration and displacement respectively.
• Maintains consistent sign conventions and units, for example taking upwards as positive so that a = −9.8 m s⁻² under gravity.

Common mistakes and how to put them right

• Treating acceleration as variable and applying constant-acceleration formulae outside their range.

  Put it right: Check that a is constant before using them, otherwise use calculus.

• Mixing up u and v in v² = u² + 2as.

  Put it right: Label initial and final velocities explicitly before substituting.

• Applying a single scalar equation to a two-dimensional problem.

  Put it right: Resolve into perpendicular components and apply the formulae to each component separately.

Exam tips

• List the known and unknown quantities with signs before choosing an equation, so the omitted variable identifies the formula to use.
• Show the derivation from a = dv/dt and v = ds/dt when a question asks you to derive rather than quote a formula.
• For vector motion, write the x- and y-equations on separate lines and keep the unit vectors throughout to avoid losing direction marks.