A Level · Cambridge (CIE) · 9709

Calculate total distance when a particle changes direction

Find every velocity zero in the stated interval, test for reversal and add the magnitudes of displacement on each motion segment.

Section 4.2 · Kinematics of motion in a straight line

Mechanics · Straight-line kinematics · 4.2

Exam scope and assessment notes

Mechanics (Paper 4, AS & A Level) · Mathematics 9709, 2026–2027, section 4.2. Motion is restricted to one dimension. Choose a positive direction; distance and speed are scalars, while displacement, velocity and acceleration carry a direction. Deceleration means decreasing speed, not necessarily negative acceleration. Calculus uses Paper 1 techniques only. Constant-acceleration formulae require constant acceleration on the stated interval; use a new interval and its initial velocity when conditions change. Where vertical motion occurs, use the supplied g = 10 m s⁻² and the stated no-air-resistance model. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Solve linear and quadratic velocity equations.
Evaluate a polynomial primitive at interval endpoints.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

For v = 4 − 2t on 0 ≤ t ≤ 3, where should the interval be split for distance?

What you will learn

Locate and verify all reversals in a given time interval.
Calculate distance and displacement separately with graph areas or integration.

For v = 6 − 2t on 0 ≤ t ≤ 5 s, reversal occurs at 3 s. The positive triangular area is 9 m and the negative triangular area is −4 m. Adding signed areas gives displacement +5 m, while adding magnitudes gives distance 13 m. Average velocity over the full 5 s is +1 m s⁻¹ and average speed is 2.6 m s⁻¹. The negative travel must contribute positively to distance.

Split the interval where velocity changes sign

Displacement is the signed integral of velocity, but total distance is the integral of |v|. A practical Paper 1 method is to solve v = 0, retain roots within the time interval, inspect signs, and evaluate displacement separately on intervals of one direction. Add the absolute values of those segment displacements. A velocity zero without a sign change is momentary rest and does not require a new direction, though splitting there is harmless.

For v = 6 − 2t on 0 ≤ t ≤ 5, velocity changes from positive to negative at 3 s. A primitive is S(t) = 6t − t². Its values are S(0) = 0, S(3) = 9 and S(5) = 5 m. Thus the segment displacements are +9 and −4 m. Their signed sum gives displacement 5 m, while their magnitudes give distance 13 m. Taking |S(5) − S(0)| would miss the four metres of negative travel.

Multiple reversals and a non-zero origin need careful bookkeeping

For v = t² − 4t + 3 = (t − 1)(t − 3), the particle reverses at both 1 and 3 s on [0, 4]. A primitive is S = t³/3 − 2t² + 3t. It has values 0, 4/3, 0 and 4/3 at 0, 1, 3 and 4 s. The three displacements are +4/3, −4/3 and +4/3 m; total distance is 4 m although displacement is only 4/3 m.

The additive initial position cancels from every segment change, so a non-zero starting coordinate does not alter this method. Once distance and displacement are known, divide each by the whole elapsed time for average speed and average velocity respectively. Include units and avoid rounding intermediate turning positions: exact roots and fractions preserve accuracy before the requested final rounding. Check that total distance is at least the magnitude of displacement.

Pause and explain

Segment displacements are +4/3, −4/3 and +4/3 m. What is total distance?

Put the idea to work

Worked example

For v = 6 − 2t m s⁻¹ on 0 ≤ t ≤ 5 s, calculate distance, displacement and both whole-journey averages.

Show the worked solution

Solve v = 0 to get reversal at 3 s; the primitive S = 6t − t² has values 0, 9 and 5 m.
The particle travels 9 m positively then 4 m negatively, giving distance 13 m and displacement +5 m.
Divide by 5 s: average speed is 2.6 m s⁻¹ and average velocity is +1 m s⁻¹.

Answer Distance 13 m; displacement +5 m; average speed 2.6 m s⁻¹; average velocity +1 m s⁻¹.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

For v = 4 − 2t m s⁻¹ over 0 ≤ t ≤ 3 s, find displacement and total distance.

Give me a hint

Evaluate the primitive at the start, reversal and end.

Compare my reasoning

The reversal is t = 2 s and a primitive is S = 4t − t².
S(0) = 0, S(2) = 4 and S(3) = 3 m, giving segment changes +4 and −1 m.
Displacement is +3 m and distance is 4 + 1 = 5 m.

Displacement +3 m; distance 5 m.

Look for these in your work

Verify the velocity sign changes at 2 s.
Add segment magnitudes for the distance.

2 · Independent

For v = t² − 4t + 3 m s⁻¹ on 0 ≤ t ≤ 4 s, find reversals, distance, displacement, average speed and average velocity.

Give me a hint

There are two roots inside the interval, creating three one-direction stages.

Compare my reasoning

Factor to get reversal times 1 and 3 s; use S = t³/3 − 2t² + 3t.
At 0, 1, 3 and 4 s, S is 0, 4/3, 0 and 4/3 m, so distance is 4 m and displacement is 4/3 m.
Over 4 s, average speed is 1 m s⁻¹ and average velocity is +1/3 m s⁻¹.

Reversals at 1 and 3 s; distance 4 m; displacement +4/3 m; average speed 1 m s⁻¹; average velocity +1/3 m s⁻¹.

Look for these in your work

Evaluate all four interval endpoints exactly.
Check distance exceeds the displacement magnitude and use total elapsed time.

3 · Transfer

A particle’s position is s = t² − 6t + 10 m for 0 ≤ t ≤ 5 s. Find its reversal, distance, displacement and both average rates.

Give me a hint

The 10 m initial coordinate cancels from changes but must be included in endpoint positions.

Compare my reasoning

Velocity is 2t − 6, so reversal occurs at 3 s.
Positions at 0, 3 and 5 s are 10, 1 and 5 m; distance is 9 + 4 = 13 m and displacement is 5 − 10 = −5 m.
Average speed is 13/5 = 2.6 m s⁻¹ and average velocity is −5/5 = −1 m s⁻¹.

Reversal 3 s; distance 13 m; displacement −5 m; average speed 2.6 m s⁻¹; average velocity −1 m s⁻¹.

Look for these in your work

Use the initial coordinate rather than assuming s(0) = 0.
Separate the path length from the negative net change.

Common mistakes

Absolute net displacement is not total distance after a reversal.
Keep only turning times within the stated interval.
Use all motion segments and postpone rounding their endpoints.

Recall without your notes

What can you explain now?

Why is |∫₀⁵ v dt| insufficient for distance when v changes sign?

Compare with the explanation

Positive and negative travel cancel in the integral before its magnitude is taken.

Distance must add the magnitude of each one-direction segment instead.

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Make it stick

Solve v = 0, inspect signs, evaluate every segment displacement and add magnitudes for distance.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.