A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Addition of Forces and Resultant Forces in Dynamics — mark scheme explained
The short answer
In AQA A-Level Mathematics, understanding the addition of forces and resultant forces is crucial for solving dynamics problems involving motion in a plane. This topic covers how to combine multiple forces acting on an object to determine the overall effect they have on its motion.
The question
A block of mass 2 kg is on a frictionless surface. Two forces act on it: one of 6 N at 30° to the horizontal and another of 8 N at 120° to the horizontal. Find the resultant force and the acceleration of the block.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For questions involving the addition of forces and resultant forces, marks are typically awarded for correctly resolving forces into components, accurately adding these components, and applying Newton's second law. Ensure you show all steps clearly and include units in your final answer.
What the command words demand
- Calculate
- Perform a mathematical operation to find a specific value.
- Determine
- Find out or establish something with certainty, often through calculation.
- Find
- Identify or discover a particular value or result.
- Resolve
- Break down a force into its horizontal and vertical components.
- Show
- Demonstrate how a given result is obtained, usually step-by-step.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate approximately 5-7 minutes per question to ensure you have enough time to resolve forces, perform calculations, and check your work.
- Resolve each force into its horizontal and vertical components:0 marks
- F x1 = 6 cos 30° ≈ 5.2 N, F y1 = 6 sin 30° = 3 N.1 mark
- F x2 = 8 cos 120° = -4 N, F y2 = 8 sin 120° ≈ 6.93 N.1 mark
- Add the horizontal and vertical components:0 marks
- F x = 5.2 - 4 = 1.2 N, F y = 3 + 6.93 ≈ 9.93 N.1 mark
- Find the magnitude of the resultant force using Pythagoras' theorem:0 marks
- R = √(1.2 2 + 9.93 2 ) ≈ 9.98 N.1 mark
- Find the direction of the resultant force using the tangent function:0 marks
- tan θ = F y / F x ≈ 9.93 / 1.2 ≈ 8.275, θ ≈ tan -1 (8.275) ≈ 83.2° to the horizontal.1 mark
- Apply Newton's second law to find the acceleration:0 marks
- a = F / m = 9.98 N / 2 kg ≈ 4.99 m/s 2 .1 mark
Final answer: Resultant force: 9.98 N at 83.2° to the horizontal; Acceleration: 4.99 m/s 2
Work through every step correctly and you earn all 6 marks.
Another worked example
A car of mass 1500 kg is being pushed by two people. One person exerts a force of 300 N at an angle of 45° to the horizontal, and the other exerts a force of 200 N at an angle of 60° to the horizontal. Find the resultant force and the acceleration of the car.
- Resolve each force into its horizontal and vertical components:0 marks
- F x1 = 300 cos 45° ≈ 212.1 N, F y1 = 300 sin 45° ≈ 212.1 N.1 mark
- F x2 = 200 cos 60° = 100 N, F y2 = 200 sin 60° ≈ 173.2 N.1 mark
- Add the horizontal and vertical components:0 marks
- F x = 212.1 + 100 = 312.1 N, F y = 212.1 + 173.2 ≈ 385.3 N.1 mark
- Find the magnitude of the resultant force using Pythagoras' theorem:0 marks
- R = √(312.1 2 + 385.3 2 ) ≈ 496.7 N.1 mark
- Find the direction of the resultant force using the tangent function:0 marks
- tan θ = F y / F x ≈ 385.3 / 312.1 ≈ 1.234, θ ≈ tan -1 (1.234) ≈ 51.0° to the horizontal.1 mark
- Apply Newton's second law to find the acceleration:0 marks
- a = F / m = 496.7 N / 1500 kg ≈ 0.331 m/s 2 .1 mark
Final answer: Resultant force: 496.7 N at 51.0° to the horizontal; Acceleration: 0.331 m/s 2
Work through every step correctly and you earn all 6 marks.
Common mistakes
Forgetting to resolve forces into components.
Why it happens: Students often try to add forces directly without breaking them down into their horizontal and vertical components, leading to incorrect results.
Fix: Always resolve forces into their horizontal and vertical components before adding them.
Incorrectly using trigonometric functions for resolving forces.
Why it happens: Students may confuse sine and cosine when finding the components of a force, leading to incorrect values.
Fix: Remember that the horizontal component is given by F cos θ and the vertical component by F sin θ .
Forgetting to apply Newton's second law separately for each direction.
Why it happens: Students sometimes try to use a single equation for both horizontal and vertical components, leading to incorrect acceleration values.
Fix: Apply F = ma separately for the horizontal and vertical directions.
Incorrectly calculating the magnitude of the resultant force.
Why it happens: Students may forget to use Pythagoras' theorem or make arithmetic errors when finding the magnitude of the resultant force.
Fix: Use R = √( F x 2 + F y 2 ) to find the magnitude of the resultant force accurately.
Incorrectly calculating the direction of the resultant force.
Why it happens: Students may make errors in using the tangent function or interpreting the angle, leading to incorrect directions.
Fix: Use tan θ = F y / F x and ensure you interpret the angle correctly based on the quadrant.
Forgetting to consider all forces acting on an object.
Why it happens: Students may overlook some forces, leading to incomplete or incorrect solutions.
Fix: Always list and consider all forces acting on the object before performing any calculations.
Incorrectly applying vector addition methods.
Why it happens: Students may use the parallelogram rule or triangle method incorrectly, leading to incorrect resultant forces.
Fix: Practice and understand both methods thoroughly. Ensure you draw vectors accurately and follow the steps correctly.
Forgetting to convert angles from degrees to radians when using trigonometric functions.
Why it happens: Students may use degree values in trigonometric functions that require radian input, leading to incorrect results.
Fix: Ensure you are using the correct mode (degrees or radians) on your calculator and convert as necessary.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Resultant Force | Resolve two forces into components, combine them, then find the resultant force and acceleration. | 6 |
| Resultant Force | Resolve two angled forces, combine them, and find the resultant force and resulting acceleration. | 6 |
| Resultant Force And Acceleration | Resolve forces into components, find the resultant, then apply Newton's second law for acceleration. | 6 |
| Resultant Force And Acceleration | Resolve two forces into components, find the resultant, then apply Newton's second law. | 6 |
| Total across these question types | 24 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.