AP Syllabus focus:
‘Use graphs or tables of position, velocity, or acceleration to estimate derivatives, identify intervals of increasing or decreasing motion, and describe when the object is speeding up or slowing down.’
Straight-line motion graphs provide rich insight into how an object moves, allowing students to infer instantaneous behavior from visual or tabular information.
Understanding Motion Through Graphical and Tabular Representations
When motion is presented graphically or in a table, connections among position, velocity, and acceleration must be inferred rather than computed symbolically. This requires interpreting slopes, concavity, and changes between data values. Students should view graphs and tables as tools for estimating how an object moves at specific moments and over intervals.
Identifying Information on Position, Velocity, and Acceleration Graphs
Graphs of different motion quantities tell different stories about the object’s movement.
Position Graphs
A position function gives the object’s location over time. On a position–time graph, the key feature to analyze is the slope, which represents the velocity at any moment.
Velocity: The instantaneous rate of change of position with respect to time; on a position graph, it is represented by the slope of the tangent line.
A position graph with a steep positive slope indicates fast forward motion, while a flat slope indicates momentary rest.
This displacement–time graph shows a straight-line position function whose slope represents constant average velocity. The labeled intervals and illustrate how slope is computed and interpreted as rate of change, mirroring the derivative concept for linear functions. The numerical values included provide a concrete example but are not essential to the underlying idea that slope represents velocity. Source.
Velocity Graphs
Velocity graphs illustrate both direction and speed directly. The sign of the velocity determines direction, and the magnitude indicates speed. Changes in velocity reveal how acceleration behaves.
Acceleration: The instantaneous rate of change of velocity with respect to time; on a velocity graph, it is represented by the slope.
A velocity graph rising upward indicates positive acceleration, while a downward slope indicates negative acceleration.
Acceleration Graphs
Although less commonly given, acceleration graphs describe how forces acting on the object influence changes in velocity. Students use the sign of acceleration to determine whether the object is speeding up or slowing down.
Estimating Derivatives from Graphs and Tables
The syllabus emphasizes using graphs and tables to estimate derivatives, which means approximating velocity from position data or acceleration from velocity data.
Using Graphs
To estimate the derivative at a point from a graph:
Draw or visualize a tangent line at the point of interest.
Approximate its slope using nearby points on the graph.
Interpret the slope in the problem’s context.
Interpret the slope in the problem’s context.
Using Tables
Tables typically provide discrete values of position or velocity. To estimate derivatives:
Use average rates of change over very small intervals to approximate an instantaneous rate.
Select values immediately surrounding the target time to refine the approximation.
Always state units clearly (e.g., meters per second, feet per second squared).
These estimates allow students to determine how fast a quantity is changing at a specific moment even when no formula is provided.
A sentence here ensures spacing before the next definition block.
Instantaneous Rate of Change: The value of the derivative at a specific point, representing how fast a quantity is changing at that exact moment.
Determining Increasing and Decreasing Motion
The syllabus requires identifying intervals of increasing or decreasing motion. These interpretations depend on whether students are analyzing position, velocity, or acceleration information.
From Position Graphs or Tables
If the slope of the position graph is positive, velocity is positive, and the object moves forward.
If the slope is negative, velocity is negative, and the object moves backward.
If the slope is zero, the object is momentarily at rest.
From Velocity Graphs or Tables
Intervals where velocity increases or decreases correspond to the sign of acceleration:
A positive slope on the velocity graph indicates increasing velocity.
A negative slope indicates decreasing velocity.
These relationships help determine how external influences cause changes in motion.
Describing When an Object Is Speeding Up or Slowing Down
A central requirement of this subsubtopic is explaining speeding up and slowing down based on graphical or tabular information. Students must compare the signs of velocity and acceleration.
Key Principles for Interpretation
An object speeds up when velocity and acceleration share the same sign.
Both positive → increasing speed in the positive direction.
Both negative → increasing speed in the negative direction.
An object slows down when velocity and acceleration have opposite signs.
Positive velocity with negative acceleration.
Negative velocity with positive acceleration.
These ideas apply whether values come from a graph or a table.
This velocity–time graph shows a skydiver whose velocity increases rapidly and then levels off at terminal velocity. The positive slope at early times corresponds to positive acceleration, while the flattening curve indicates acceleration approaching zero. The drag-based model that produces this shape includes extra physical detail, but its slope behavior directly illustrates speeding up and then moving at nearly constant speed. Source.
A normal sentence here separates conceptual text from the next equation block.
= instantaneous velocity (units of position per unit time)
= position function (units of length)
= instantaneous acceleration (units of velocity per unit time)
= velocity function
Applying Concepts Across Representations
Students should be comfortable transitioning between graphs and tables and understanding how estimated slopes reveal instantaneous behavior. Interpretation always depends on context, requiring careful attention to units, direction, and whether the object is moving faster, slower, forward, or backward.
FAQ
A change in direction occurs when the velocity crosses zero on a velocity–time graph. At this point, the object momentarily stops before reversing direction.
When interpreting acceleration around a direction change, focus on the sign of the acceleration relative to the sign of the new velocity direction. The object may be speeding up after the reversal even if it was slowing down just before it.
Use a small, symmetric interval around the point of interest.
• Choose two points equally spaced on either side of the desired time.
• Draw or visualise a secant line through these two points.
• Use its slope as your estimate.
This method reduces error from irregular graph sketching and often produces a more stable approximation than a single-sided estimate.
Prefer the smallest available interval around the point where the derivative is required. This best mimics the idea of an instantaneous rate.
If values are unevenly spaced, use the closest times on either side. One-sided estimates are acceptable but may be less accurate. When two-sided data exist, favour a symmetric difference, as it balances errors from local fluctuations.
Concavity relates to how the graph bends and provides insight into acceleration.
• A graph bending upwards (shaped like a cup) indicates positive acceleration.
• A graph bending downwards indicates negative acceleration.
• A change in concavity signals a change in acceleration direction.
This approach relies solely on visual interpretation and does not require any explicit formula.
Estimation depends on how each student chooses nearby points, draws tangents, or interprets the curve’s steepness.
Small differences are expected because graphical estimation is inherently approximate. Discrepancies should be checked for reasonableness:
• Are the signs consistent?
• Are the magnitudes within a plausible range?
• Does the behaviour match the graph’s general trend?
Marks on exam questions typically reward correct reasoning more than exact numerical agreement.
Practice Questions
Question 1 (1–3 marks)
A particle moves along a straight line, and its position–time graph is shown to be increasing with a decreasing slope over the interval 0 ≤ t ≤ 4.
Based on the graph, determine whether the particle is speeding up or slowing down on this interval and justify your answer.
Question 1
• 1 mark: Correctly states that the particle is slowing down.
• 1 mark: Identifies that the slope of the position–time graph (the velocity) is decreasing.
• 1 mark: Explains that decreasing slope means velocity is getting smaller in magnitude, so the particle slows down.
Question 2 (4–6 marks)
A table of velocity values for a moving object is given below.
t (seconds): 0 1 2 3 4
v (m/s): 2 5 6 5 3
(a) Estimate the object’s acceleration at t = 2 seconds using the values in the table.
(b) State whether the object is speeding up or slowing down at t = 2 seconds and justify your answer using both the estimated acceleration and the velocity value.
(c) Describe one interval on which the object’s velocity is increasing and one interval on which it is decreasing.
Question 2
(a)
• 1 mark: Uses values surrounding t = 2 (e.g., v(3) and v(1)) to estimate acceleration as an average rate of change.
• 1 mark: Computes a correct estimate: for example, acceleration ≈ (5 − 5) / (3 − 1) = 0, or uses a different valid pair (e.g., forward difference (5 − 6)/(3 − 2) = −1 or backward difference (6 − 5)/(2 − 1) = 1). Any reasonable estimate earns credit.
(b)
• 1 mark: Correctly states whether the object is speeding up or slowing down at t = 2 (answers depend on chosen acceleration estimate).
• 1 mark: Justifies using both velocity at t = 2 (which is positive) and the sign of the estimated acceleration.
For example: If acceleration is estimated as negative, object is slowing down since velocity is positive and acceleration is negative.
If acceleration is estimated as positive, object is speeding up since both velocity and acceleration are positive.
If acceleration is estimated as zero, motion is neither speeding up nor slowing down at that instant.
(c)
• 1 mark: Identifies an interval where velocity is increasing, such as 0 ≤ t ≤ 2 (any strictly increasing portion earns the mark).
• 1 mark: Identifies an interval where velocity is decreasing, such as 2 ≤ t ≤ 4.
