Kinematics in Additional Math is about motion in a straight line using calculus, velocity-time graphs and the relationships between displacement, velocity and acceleration. The reports show that the most common errors involve the direction of the calculus, signs, and what a graph represents. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Differentiating and integrating. Not realising that acceleration is found by differentiating the expression for velocity, integrating an exponential function with a sign error or by differentiating instead, and in one question, finding the time for which s is stationary when the question was about v. Others did not differentiate correctly (for example giving v = 4te^(4t−1) − 2te^(2t−1)), or integrated ∫2e^t dt as t ln(2e).
- Constants. Omitting the constant of integration or making an algebraic slip when calculating it, and an incorrect constant in the second expression when finding displacements at t = 1 and t = 3.
- Distance, displacement, velocity and speed. Confusion between displacement and distance (−10 was a common answer), and between velocity and speed, with −4.91 m/s given where 4.91 m/s was needed. In another question, students found x₂ − x₀ = 18 − 8 = 10 or just stated x₂ = 18.
- Acceleration. Candidates did not appreciate that a negative gradient means the acceleration should also be negative, or wrongly assumed that the acceleration should be positive (1.37 instead of −1.37 m/s²). Others found that the acceleration at t = 0 is non-zero.
- Velocity-time graphs. Omitting to mark any scale or making a sign error when marking −8, leaving V off the vertical axis, showing the final horizontal segment as positive instead of negative, and drawing curves in part (a) which then did not justify constant acceleration in a later part. Many attempted a graph with connected ruled sections, and a few drew only two correct sections.
- Areas under the graph. Including only the triangular part of a trapezium and not the rectangular part, or trying to find the area of a big triangle from 20 to 40. One question needed students to find the gradient of the line at t = 40.
- Using constant acceleration formulae. Attempting suvat equations where they did not apply and often making little progress, and misinterpreting the information about the total distance (forming ½TV = 1104, for example).
- Calculator settings. Having the calculator in degrees instead of radians when evaluating trigonometric functions of time.
- Exact answers. Converting ln 2 to a decimal and giving an inexact answer, and leaving an answer such as 16e^(4 ln 2) − 40e^(2 ln 2) unfinished.
- Quotient and product rule. Errors in the numerator of the quotient rule, and not rewriting the original equation as a product before using the product rule.
- Slips. Writing (t + 3) instead of (t + 1) through carelessness, and dividing or multiplying by 35 instead of dividing by −15.
What examiners advise
- Draw a diagram or graph. The reports say a diagram, while not always needed for full credit, is good practice and should be encouraged.
- Use calculus when velocity or acceleration is given as a function of time, and check that you are differentiating or integrating the right way.
- Remember the meaning: the gradient of a velocity-time graph is the acceleration, and the area under it is the displacement.
- Label graphs with scales and the quantity on each axis, and show the correct sign for each segment.
- Check the calculator mode (radians) for trigonometric functions of time.
- Be guided by the mark allocation to judge how much work is needed.
Habits that help
- Write "s → v → a" with arrows for differentiation going right and integration going left.
- At the end of a question, check that the sign of your answer fits the direction of motion.
- For a graph, list the key times and velocities in a short table before you draw it.
Questions students ask
When do I use calculus in Additional Math kinematics?
When velocity or acceleration is given as a function of time. Suvat equations do not apply then, and students who attempted them made little progress. Velocity is found by integrating acceleration, and acceleration by differentiating velocity.
What do the gradient and area of a velocity-time graph mean?
The gradient is the acceleration and the area under it is the displacement. Candidates forgot that a negative gradient means negative acceleration. Include the rectangular as well as the triangular part of a trapezium.
What is the difference between displacement and distance, and between velocity and speed?
Displacement and velocity have signs, distance and speed do not. Reports record −10 given as a distance and −4.91 m/s given where a speed was wanted.
How should I label a velocity-time graph?
With scales and the quantity on each axis (include V on the vertical axis), and the correct sign for each segment. Do not show the final horizontal segment as positive when it is negative.
Why should I check the calculator mode?
For trigonometric functions of time the calculator must be in radians. Students had it in degrees.
Should I leave answers exact?
Yes where required. Converting ln 2 to a decimal gave an inexact answer, and leaving 16e^(4 ln 2) − 40e^(2 ln 2) unfinished lost marks.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.