A Level · Cambridge (CIE) · 9709

Conserve mechanical energy on smooth slopes and slides

Exchange kinetic and gravitational potential energy in a loss-free model, including a curved slide whose speed depends on height change.

Section 4.5 · Energy, work and power

Mechanics · Energy, work and power · 4.5

Exam scope and assessment notes

Mechanics (Paper 4, AS & A Level) · Mathematics 9709, 2026–2027, section 4.5. Use g = 10 m s⁻² throughout these problems. Masses are positive and constant. Work by a constant force is Fd cos θ, with θ between force and displacement; scalar-product notation is not required. Kinetic energy is ½mv² and gravitational potential-energy change is mgΔh. State the system and count gravitational work or potential-energy change once, not both. Mechanical energy is conserved only under the stated loss-free model; friction and supplied resistance transfer mechanical energy to other forms. Smooth curved-slide examples use overall energy changes and assume contact throughout, rather than requiring a normal-reaction calculation. Average power is work divided by elapsed time; P = Fv applies to a force along the motion. Constant driving power generally gives variable force and acceleration; instantaneous calculations do not justify constant-acceleration formulae over a whole journey. Check string, contact and motion-state changes before continuing a model. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Use kinetic energy and gravitational height changes.
State the assumptions required for a smooth fixed path.

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

Quick readiness check

When may K + U be conserved on a fixed path?

What you will learn

Use K + U conservation when no supplied force removes mechanical energy.
Calculate endpoint speed on a smooth curved slide using only vertical height change.

A particle starts from rest on a fixed smooth curved slide with vertical drop 5 m. Contact is maintained and resistance is neglected. Gravitational potential energy transfers into kinetic energy, giving endpoint speed 10 m s⁻¹ for g = 10 m s⁻². The track shape is schematic; the marked vertical drop, rather than curved path length, enters the overall energy calculation. No claim about normal reaction or travel time is made.

A loss-free model exchanges kinetic and potential energy

For a smooth fixed slope with no supplied resistance or drive, gravity transfers energy between K and U while the normal reaction does no work. Thus ½mu² + mgh₁ = ½mv² + mgh₂. A particle released from rest and dropping vertically by 5 m reaches speed √(2 × 10 × 5) = 10 m s⁻¹. Positive mass cancels, so the predicted speed is independent of mass under these assumptions.

A particle entering a smooth descent at speed 4 m s⁻¹ and dropping 2 m has v² = 4² + 2 × 10 × 2 = 56, so v = 2√14 m s⁻¹. That is an endpoint speed; it does not by itself provide travel time or acceleration at every point. Mechanical energy need not be conserved when a drive, friction, resistance or impact changes the balance.

A curved path uses its overall height change

A particle sliding on a fixed smooth curved track, with contact stipulated throughout and no resistance, obeys the same endpoint energy relation. The curved shape changes local forces and directions, but the gravitational potential change depends only on vertical height. We do not need to compute a normal reaction or a curved-motion acceleration to answer the endpoint-speed question in this section.

If a particle starts at speed 8 m s⁻¹ and climbs a smooth path, its greatest possible height increase before speed becomes zero is 8²/(2 × 10) = 3.2 m. An endpoint 4 m higher would require v² = 64 − 80 < 0 and cannot be reached under that loss-free initial-energy model. This is an energy bound, subject to the stated contact and path assumptions, rather than a claim about a real track’s safety or full trajectory.

Pause and explain

For a smooth curved slide, which information determines endpoint speed in the loss-free model?

Put the idea to work

Worked example

A particle is released from rest on a fixed smooth curved slide with a 5 m vertical drop to its endpoint. Contact is maintained and resistance is neglected. Find endpoint speed.

Show the worked solution

The potential-energy decrease is 5mg.
By mechanical-energy conservation, ½mv² = 5mg.
Cancel positive m: v² = 2 × 10 × 5 = 100, so endpoint speed is 10 m s⁻¹.

Answer Endpoint speed 10 m s⁻¹, independent of mass in this model.

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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

A particle is released from rest on a smooth fixed slide with vertical drop 5 m. Contact is maintained and resistance is neglected. Find its endpoint speed.

Give me a hint

Equate lost potential energy to gained kinetic energy.

Compare my reasoning

The decrease in potential energy is 5mg.
The energy equation is ½mv² = 5mg.
Thus v² = 100 and speed is 10 m s⁻¹.

Endpoint speed 10 m s⁻¹.

Look for these in your work

Use the supplied vertical drop.
State the loss-free smooth-contact assumptions.

2 · Independent

A particle enters a smooth fixed descent at speed 4 m s⁻¹ and drops vertically by 2 m. With contact maintained and no resistance, find endpoint speed.

Give me a hint

Initial kinetic energy is non-zero and must be retained.

Compare my reasoning

Energy gives ½mv² = ½m(4²) + mg(2).
Cancel mass to obtain v² = 16 + 40 = 56.
Endpoint speed is √56 = 2√14 m s⁻¹.

Endpoint speed 2√14 m s⁻¹.

Look for these in your work

Include initial kinetic energy.
Use the non-negative speed root.

3 · Transfer

A particle starts up a smooth fixed path at speed 8 m s⁻¹, with no driving force or resistance and contact maintained while it moves. Find the maximum possible vertical rise and decide whether it can reach a point 4 m higher.

Give me a hint

At the height limit, kinetic energy is zero; a higher point would require negative kinetic energy.

Compare my reasoning

The limiting equation is ½m(8²) = mgΔh, so Δh = 64/20 = 3.2 m.
At rise 4 m, the predicted squared speed would be 64 − 80 = −16.
That is impossible, so the point cannot be reached under the stated energy model.

Maximum vertical rise 3.2 m; a point 4 m higher is unreachable in this model.

Look for these in your work

Use zero final speed at the energy limit.
Reject a negative squared-speed prediction rather than changing its sign.

Common mistakes

Use vertical drop, not curved path length.
Energy does not by itself supply the travel time.
Contact and no-loss assumptions must be stated.

Recall without your notes

What can you explain now?

Does the curved path’s total length replace vertical drop in mgΔh?

Compare with the explanation

No; gravitational potential change depends on vertical height change.

Other path-dependent work must be supplied separately if friction or resistance acts.

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Make it stick

Under a stated loss-free smooth model, exchange K and U using vertical height differences and check the available-energy bound.

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.