A Level · Cambridge (CIE) · 9709

Find instantaneous acceleration of a powered vehicle on a hill

Combine P/v with resistance and the slope weight component, distinguish instantaneous acceleration from cruise speed and avoid a constant-acceleration assumption.

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 F = ma and the along-slope gravitational component.
Use instantaneous P = Fv and convert kilowatts to watts.

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

Quick readiness check

A 40 kW drive acts at uphill speed 20 m s⁻¹. What is its driving force?

What you will learn

Find driving force and instantaneous acceleration at a specified positive uphill speed.
Calculate a possible steady speed and interpret speeding up or slowing down relative to it.

Power supplies a drive force at the current speed

A 1000 kg vehicle moves uphill at 20 m s⁻¹ with driving power 40 kW, resistance 500 N and sin θ = 1/20. Its drive force at this instant is 40 000/20 = 2000 N. The downslope weight component is 1000 × 10 × 1/20 = 500 N. With up the hill positive, 2000 − 500 − 500 = 1000a, so a = +1 m s⁻². The normal reaction does no along-path work under the fixed-road model.

At a larger speed with the same power and constant resistance, driving force is smaller and the instantaneous acceleration changes. At speed 50, the drive is 800 N while total opposition remains 1000 N, giving a = −0.2 m s⁻². The vehicle still moves uphill but slows. The acceleration sign does not immediately reverse its current direction of travel.

A steady speed is a force balance, not a whole-journey average

For this example, zero acceleration requires P/v = 500 + 500 = 1000 N, so possible steady speed is v = 40 000/1000 = 40 m s⁻¹. Speeds below 40 have positive acceleration and speeds above 40 have negative acceleration while the same model holds. The value is a model prediction, not permission to infer travel time or a real safe road speed.

Constant power does not make acceleration constant over an interval because P/v changes with v. Do not substitute the acceleration at 20 into v = u + at for an entire speed-changing journey. A downhill problem needs its own signed gravitational component, and a speed-dependent resistance must be evaluated from its supplied law. Our numerical examples use specified positive uphill speeds and constant opposition; a zero or negative denominator needs a different interpretation, not an automatic finite cruise speed.

Pause and explain

In the 1000 kg hill example, speed 50 m s⁻¹ gives drive 800 N and total opposition 1000 N. What follows?

Put the idea to work

Worked example

A 1000 kg vehicle moves uphill at 20 m s⁻¹, with power 40 kW, resistance 500 N and sin θ = 1/20. Find its instantaneous acceleration and possible steady speed in this model.

Show the worked solution

At the stated speed, drive force is 40 000/20 = 2000 N and downslope weight component is 500 N.
Newton’s law gives a = (2000 − 500 − 500)/1000 = 1 m s⁻² uphill.
For zero acceleration, v = 40 000/(500 + 500) = 40 m s⁻¹, provided the model remains valid.

Answer Instantaneous a = +1 m s⁻²; possible steady speed 40 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

A 1000 kg vehicle moves uphill at 20 m s⁻¹ with drive power 40 kW, resistance 500 N and sin θ = 1/20. Find its instantaneous acceleration.

Give me a hint

Calculate the drive from the current speed and include both opposing forces.

Compare my reasoning

Drive force is 40 000/20 = 2000 N.
The slope weight component is 1000 × 10 × 1/20 = 500 N, so resultant is 2000 − 500 − 500 = 1000 N uphill.
Acceleration is 1000/1000 = +1 m s⁻² at this instant.

Instantaneous a = +1 m s⁻² uphill.

Look for these in your work

Convert kilowatts and use the instantaneous speed.
Subtract resistance and the slope weight component separately.

2 · Independent

A 1200 kg vehicle moves uphill at 15 m s⁻¹ with power 30 kW, resistance 400 N and sin θ = 1/20. Find its instantaneous acceleration and possible steady speed under this constant-resistance model.

Give me a hint

The steady-speed condition sets resultant force to zero at a potentially different speed.

Compare my reasoning

At 15, drive is 30 000/15 = 2000 N and slope weight component is 1200 × 10/20 = 600 N.
Instantaneous acceleration is (2000 − 400 − 600)/1200 = 5/6 m s⁻² uphill.
For zero acceleration, drive equals 1000 N, so possible steady speed is 30 000/1000 = 30 m s⁻¹.

Instantaneous a = 5/6 m s⁻²; possible steady speed 30 m s⁻¹.

Look for these in your work

Use total opposing force in both calculations.
Do not equate the stated current speed with the predicted steady speed.

3 · Transfer

The 1000 kg vehicle with constant power 40 kW, resistance 500 N and sin θ = 1/20 is moving uphill at 50 m s⁻¹. Find its instantaneous acceleration, explain whether it is speeding up and decide whether that acceleration can be used for the whole later journey.

Give me a hint

Compare current drive P/v with the unchanged total opposing force.

Compare my reasoning

Current drive is 40 000/50 = 800 N while total opposition is 1000 N.
Thus a = (800 − 1000)/1000 = −0.2 m s⁻²; its positive uphill velocity is decreasing in magnitude.
As speed changes, P/v and acceleration change, so −0.2 is an instantaneous value and cannot be assumed constant over the whole later journey.

Instantaneous a = −0.2 m s⁻²; uphill speed decreases; whole-journey constant acceleration is not justified.

Look for these in your work

Keep the current velocity direction distinct from acceleration direction.
State the instantaneous limitation of the constant-power force calculation.

Common mistakes

Subtract both resistance and the uphill gravitational opposition.
An instantaneous acceleration is not constant over a whole constant-power journey.
A negative acceleration can mean slowing while still moving uphill.

Recall without your notes

What can you explain now?

May the instantaneous acceleration at one speed be used as constant throughout a speed-changing constant-power journey?

Compare with the explanation

No; the driving force P/v changes with speed, so acceleration generally changes.

Use a separately justified interval model if calculating elapsed travel time.

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Make it stick

At the specified positive speed, use P/v for drive, subtract all signed opposition and divide the resultant by mass.

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.