Recognise changing speed on curved distance-time graphs
Use steepness to distinguish accelerating and decelerating motion without treating distance as speed.
Section P1.2 · Motion
Exam scope and assessment notes
Mapped to the named 0973 section using teaching from the corresponding single-science course. A 0973-specific review of every Core and Supplement objective is still in progress. Written practice is self-checked; it does not demonstrate supervised laboratory competence or award an official Cambridge grade. This qualification reports two grades on the 9–1 scale; confirm the entry route for your zone and exam series.
Before you begin
- Connect a straight distance–time gradient to speed.
- Recognise a curve's local steepness.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does distance–time steepness describe?
What you will learn
- Interpret changing gradients qualitatively.
- Distinguish a flattening line from a falling distance.
Steeper means faster on these axes
A rising cumulative-distance curve that becomes steeper shows increasing speed. One that rises but becomes less steep shows decreasing speed. The distance can keep increasing while speed decreases, because the object is still travelling forward along its path. A truly horizontal part indicates rest.
For the example data at t = 0, 1, 2, 3, 4 s, cumulative distances 0, 1, 4, 9, 16 m increase by 1, 3, 5 and 7 m in equal one-second intervals. The interval average speeds increase. A smooth curve consistent with s = t² becomes progressively steeper; the original five measurements alone do not prove an exact law at every unmeasured instant.
Decreasing increments do not mean negative distance
A contrasting set of distances 0, 7, 12, 15, 16 m at the same times increases by 7, 5, 3 and 1 m. Its slope becomes smaller, indicating deceleration over the intervals. A smooth illustrative curve s = 8t − t² rises and flattens over zero to four seconds.
Sketch a qualitative curve with time increasing rightward, and label whether its gradient is increasing, constant or decreasing. Do not calculate a section's speed as its final distance divided by final time unless the section starts at the origin and that average is the requested quantity. Core requires qualitative curve interpretation; numerical acceleration is taught in Supplement lessons.
Sources for this lesson
Pause and explain
A rising distance curve gets less steep. What is happening?
Worked example
Cumulative distances at one-second intervals are 0, 7, 12, 15 and 16 m. Describe the speed trend.
Show the worked solution
- Distance increments are 7, 5, 3 and 1 m.
- The intervals all last one second.
- Interval speeds decrease, so the motion is decelerating.
Answer Speed decreases while cumulative distance still increases.
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Test this topic →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.
At times 0,1,2,3,4 s, distances are 0,1,4,9,16 m. Describe the speed trend.
Give me a hint
Find distance increments across equal time intervals.
Compare my reasoning
- Increments are 1,3,5,7 m.
- Each interval lasts one second.
- Interval speeds increase; the curve becomes steeper.
Accelerating motion; increasing interval speeds.
Look for these in your work
- Use equal-time comparisons.
- Connect to increasing slope.
At the same times, distances are 0,7,12,15,16 m. Describe the trend and whether distance decreases.
Give me a hint
Read increments rather than comparing a distance reading with a speed.
Compare my reasoning
- Increments are 7,5,3,1 m.
- The interval speeds decrease.
- Distance still increases throughout these intervals.
Decelerating; cumulative distance still increases.
Look for these in your work
- Identify decreasing slope.
- Keep distance and speed distinct.
Sketch a distance–time journey that accelerates, then maintains constant speed, then slows to rest. Explain the final slope.
Give me a hint
Join an increasingly steep curve to a straight section and then a flattening curve.
Compare my reasoning
- The first rising curve becomes steeper.
- The constant-speed part is a straight rising line.
- The final curve flattens to zero gradient as motion stops.
Steepening curve, straight rise, then flattening to a horizontal tangent.
Look for these in your work
- Match each slope pattern.
- Keep cumulative distance non-decreasing.
Common mistakes
- A flattening rising curve does not mean distance decreases.
- A high distance reading does not itself mean high speed.
- A few plotted points do not establish all intervening motion.
What can you explain now?
If a distance–time curve becomes steeper, what happens to speed?
Compare with the explanation
It increases.
More distance is added per unit time as the local gradient grows.
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Make it stick
Sketch accelerating and decelerating cumulative-distance curves and explain why both can rise.
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.