A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Calculus in Kinematics for Motion in a Straight Line and 2D Vectors — mark scheme explained
The short answer
In AQA A-Level Mathematics, the use of calculus in kinematics is crucial for understanding motion in both one-dimensional (1D) and two-dimensional (2D) contexts. This topic involves using derivatives and integrals to describe the relationships between displacement ( r ), velocity ( v ), and acceleration ( a ).
The question
A particle moves along a straight line with its position given by the function r(t) = t 3 - 6t 2 + 9t + 5. Find the velocity and acceleration of the particle at time t = 2. [Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Find the velocity by differentiating the position function: v(t) = dr/dt = 3t 2 - 12t + 9.
- S2
Evaluate the velocity at t = 2: v(2) = 3(2) 2 - 12(2) + 9 = 12 - 24 + 9 = -3 m/s.
- S3
Find the acceleration by differentiating the velocity function: a(t) = dv/dt = 6t - 12.
- S4
Evaluate the acceleration at t = 2: a(2) = 6(2) - 12 = 12 - 12 = 0 m/s 2 .
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Find the velocity by differentiating the position function: v(t) = dr/dt = 3t 2 - 12t + 9.
- S2
Evaluate the velocity at t = 2: v(2) = 3(2) 2 - 12(2) + 9 = 12 - 24 + 9 = -3 m/s.
- S3
Find the acceleration by differentiating the velocity function: a(t) = dv/dt = 6t - 12.
- S4
Evaluate the acceleration at t = 2: a(2) = 6(2) - 12 = 12 - 12 = 0 m/s 2 .
Final answer: v(2) = -3 m/s, a(2) = 0 m/s 2
Common mistakes
- Confusing velocity with acceleration. — Always remember that velocity is the first derivative of displacement, and acceleration is the second derivative of displacement or the first derivative of velocity.
- Forgetting to include constants of integration. — Always include a constant of integration when performing indefinite integrals and use initial conditions to determine its value if necessary.
- Incorrectly applying vector operations. — Practice differentiating and integrating each component of a vector separately and then combining them into the final vector form.
- Using incorrect units. — Always check your units to ensure they are consistent with the problem statement and the physical quantities involved.
- Failing to identify whether the problem involves 1D or 2D motion. — Always start by identifying whether the problem involves one-dimensional or two-dimensional motion and use the appropriate methods accordingly.
Where the marks go
- Full worked solution (all marking points)4 marks