Q: Kinematics

Overview

Specification reference: AQA A-level Mathematics 7357, section 3.18 Q: Kinematics, Q1–Q5. Kinematics is the mathematics of describing motion without first explaining its cause. In AQA Paper 2, it can appear alone or inside a multi-step mechanics problem involving vectors, trigonometry, forces or calculus. Paper 2 lasts 2 hours for 100 marks, so candidates should budget approximately 1.2 minutes per mark and show a coherent model, not merely obtain a calculator answer.

This topic moves from precise language, through graphs and constant-acceleration formulae, to calculus and projectile motion. The unifying idea is simple: choose a positive direction, define your variables, then make every sign agree with that choice. Marks are commonly awarded for a correct model, an appropriate equation, valid substitution and a conclusion with units. A diagram, a variable table and a final sentence are therefore productive use of time.

Key Concept 1: Language, direction and units

Position is where an object is relative to an origin. Displacement is the signed change in position, while distance travelled is the total path length and cannot be negative. Velocity is the signed rate of change of displacement; speed is its non-negative magnitude. Acceleration is the rate of change of velocity. These distinctions matter most when a particle reverses direction: its displacement can be zero even though it has travelled a substantial distance.

Before using any equation, write a one-line convention such as: take upwards as positive. For vertical motion, gravity is then (a=-9.8\ \mathrm{m\ s^{-2}}). If downwards is positive, write (a=+9.8\ \mathrm{m\ s^{-2}}). Either is valid. Credit is given for consistency, not for a preferred direction. Use SI units: metres (m), seconds (s), metres per second (m s⁻¹) and metres per second squared (m s⁻²). Convert kilometres per hour to metres per second by dividing by 3.6.

Exam checkpoint

A negative velocity means the particle is moving in the chosen negative direction. A negative acceleration does not automatically mean it is slowing down. It slows only when velocity and acceleration have opposite signs.

Key Concept 2: Reading kinematics graphs

For a displacement–time graph, the gradient is velocity. A straight line has constant velocity. A horizontal line means the particle is stationary. On a curve, the gradient of a tangent gives instantaneous velocity, not the gradient of a chord drawn across a time interval.

For a velocity–time graph, the gradient is acceleration and the signed area between the graph and the time axis is displacement. Area above the axis is positive; area below is negative. To find total distance travelled, add the magnitudes of all areas, splitting the graph at every crossing of the time axis. Candidates lose easy marks when they write “area = distance” without checking whether velocity changes sign.

A useful chain is (s \rightarrow v \rightarrow a): differentiate displacement to obtain velocity, then differentiate velocity to obtain acceleration. Integrating moves in the reverse direction. This is why a displacement–time curve getting steeper has increasing velocity, and a velocity–time line with positive gradient has positive acceleration.

Key Concept 3: Constant acceleration and SUVAT

The five constant-acceleration equations apply only when acceleration is constant and the motion is one-dimensional along your chosen axis. The AQA Formulae booklet supplies these equations, but candidates must select and apply them correctly.

Here (u) is initial velocity, (v) final velocity, (a) constant acceleration, (s) displacement and (t) time. SUVAT is not a formula hunt. First list the values you know, including signs, then choose an equation containing the three known quantities and the one required quantity, while avoiding a variable you do not know. If a question says acceleration varies with time or gives (s(t)), use calculus instead.

Key Concept 4: Calculus with motion

For rectilinear motion, if displacement from the origin is (s(t)), then

[v=\frac{ds}{dt}, \qquad a=\frac{dv}{dt}=\frac{d^2s}{dt^2}.]

Conversely,

[v=\int a,dt, \qquad s=\int v,dt.]

The derivative relationships and the need for an integration constant must be recalled and used accurately. After integrating, use the initial condition immediately. For example, if (a=6t-12) and (v(0)=5), then (v=3t^2-12t+5), not simply (3t^2-12t). When asked for distance travelled, solve (v=0) to find each change of direction, calculate the displacement at those times, and add the positive lengths of the separate journeys.

Key Concept 5: Two dimensions and projectile motion

A projectile model assumes gravity is the only acceleration, air resistance is negligible, and motion is in a vertical plane. Choose horizontal (x) and vertical (y) axes. Resolve the initial velocity (u) at angle (\theta) above the horizontal into (u_x=u\cos\theta) and (u_y=u\sin\theta). The components share the same time but are otherwise independent.

Horizontally, (a_x=0), so (x=(u\cos\theta)t). Vertically, with up positive, (a_y=-g), so (y=y_0+(u\sin\theta$)t-\tfrac12gt^2). Solve the vertical equation first when the question asks when or where the particle lands. Reject a negative time as physically irrelevant. Then substitute the positive time into the horizontal equation to find range. At maximum height, the vertical component of velocity is zero, but the horizontal component generally remains non-zero.$

Mathematical relationships: formula status and selection

• (v=ds/dt), (a=dv/dt=d^2s/dt^2): Must recall and use for variable acceleration.
• (\Delta v=\int a,dt) and (\Delta s=\int v,dt): Must recall and use, including constants of integration or stated limits.
• (u_x=u\cos\theta), (u_y=u\sin\theta): Must construct using trigonometry.
• (x=x_0+u_xt) when (a_x=0): Derived from the supplied constant-acceleration equations.
• (y=y_0+u_yt-\tfrac12gt^2) for an upward-positive projectile:Derived from the supplied constant-acceleration equations.

Use (g=9.8\ \mathrm{m\ s^{-2}}) only when the question instructs that value or supplies an equivalent value. Do not round intermediate values prematurely. Keep at least three or four significant figures until the final line, then use the precision requested by the question.

Graph and data skills that earn marks

Label both axes with the quantity and unit. State a gradient calculation as a fraction with units, for example ((12-0)/(4-0)=3\ \mathrm{m\ s^{-2}}). Identify the relevant geometrical area before calculating it, and give its unit as metres when it represents displacement. On a curve, draw a tangent rather than reading a gradient from the curve by eye. If a graph crosses the axis, write separate signed areas and then decide whether the command word requires displacement or distance.

Practical applications and model limits

Kinematics underpins vehicle braking, sports trajectories, animation and navigation systems. In an exam, these contexts are models rather than practical experiments. State a sensible assumption when invited to assess a model: for instance, “air resistance is neglected, so horizontal velocity is taken as constant”. AQA A-level Mathematics has no required practical for Q: Kinematics. Instead, exam questions test modelling choices, clear diagrams and mathematical justification.

A high-mark routine

1. Sketch the path or graph and choose positive axes.
2. Record a variable table, including units and signs.
3. State the model: constant acceleration, variable acceleration, or projectile.
4. Select, derive or integrate the relevant relation and show substitution.
5. Interpret the numerical result in context, with units and an appropriate physical root.

Memory hook: Direction first, data second, equation third. This avoids the most common sign and formula-selection errors.

Checking an answer like an examiner

Before moving on, run three quick checks. First, units.In (v=u+at), acceleration multiplied by time must have the same unit as velocity. In a velocity–time graph, ((\mathrm{m\ s^{-1}})(\mathrm{s})) gives metres, which confirms why area represents displacement. A unit check will often expose an accidental mix of kilometres per hour and metres per second.

Second, sign and size. If you choose right or up as positive, a negative displacement is a location in the opposite direction, not necessarily an error. In contrast, a negative time for a landing event is normally rejected because the event is being modelled after (t=0). Ask whether the result matches the sketch: a braking distance should be positive if the direction of travel is positive, while an object projected upwards should initially rise when its vertical initial velocity is positive.

Third, conditions. Check that a constant-acceleration formula was not used on a changing acceleration, and that a projectile solution used vertical components for height and horizontal components for range. When a quadratic produces two roots, substitute or interpret both roots rather than automatically taking the first calculator value. In longer questions, a brief statement such as “the negative root is rejected since it is before projection” turns correct mathematics into a fully communicated answer.

Independent revision routine

Spend ten minutes with the formula sheet closed: redraw the two graph rules, write the five SUVAT equations, and create one example in which velocity changes sign. Then spend ten minutes on a projectile, forcing yourself to label (u\cos\theta), (u\sin\theta), (a_x=0) and (a_y=-g) before any substitution. Finish with the retrieval cues below. This alternates recall, application and explanation, which is far more exam-effective than repeatedly reading a worked solution.

Listen and recall

Use the podcast once with the guide open, then a second time with the guide closed. Pause at the quick-fire recall section and answer aloud before the response is given. This is deliberately active practice: it reveals whether a relationship is retrievable under time pressure.

Source basis

This guide follows AQA 7357 section 3.18 (Q1–Q5). AQA’s Paper 2 assessment is a 2-hour, 100-mark paper. The five constant-acceleration equations are supplied in AQA’s Formulae for A-level Mathematics booklet.