A-level Further Maths Year 12 · Topic 6
Further mechanics
Further mechanics models collisions, springs and machines.
We will use momentum and impulse, work, energy and power, and Hooke’s law for strings and springs.
Then we model collisions with the coefficient of restitution and check formulae with dimensions.
- 5 subtopics
- 260 questions
- Free: 6.1 notes and 10 questions
In this topic
- 6.1Momentum and impulse, including in vector formFree below · 51 questions
- 6.2Work, energy and power with variable resistanceOpen in Full Marks · 53 questions
- 6.3Elastic strings and springs: Hooke's law and elastic potential energyOpen in Full Marks · 53 questions
- 6.4Collisions: the coefficient of restitution and Newton's experimental lawOpen in Full Marks · 52 questions
- 6.5Dimensional analysisOpen in Full Marks · 51 questions
6.1 · Free notes
Momentum and impulse, including in vector form
Key points
- Momentum = mv (kg m s⁻¹), a vector. Impulse I = change in momentum = mv − mu; for a constant force, I = Ft (N s).
- Conservation of linear momentum: if no external impulse acts on a system, total momentum before = total momentum after.
- In two dimensions, use vectors: I = m(v − u) with v and u as vectors; conserve momentum in each direction separately.
- Impulse from a variable force is the area under a force–time graph (∫F dt).
In more detail
- Worked example: a 0.5 kg ball moving at (6i − 2j) m/s is hit and moves at (−4i + 3j) m/s. Impulse = 0.5((−4i + 3j) − (6i − 2j)) = (−5i + 2.5j) N s, magnitude ≈ 5.59 N s.
- Direction matters: take a positive direction and give velocities in the opposite direction a negative sign.
- Exam tip: draw before-and-after diagrams with masses and velocities labelled.
Practice
Momentum and impulse, including in vector form: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What is the impulse needed to change the velocity of a 2 kg mass from 3 m/s to 8 m/s?
- 22 N s
- 16 N s
- 10 N s
- 5 N s
Show answer
C: 10 N s. 2(8 − 3).
2. A force of 20 N acts for 0.5 s. What is the impulse?
- 10 N s
- 40 N s
- 20 N s
- 0.025 N s
Show answer
A: 10 N s. I = Ft.
3. When is momentum conserved?
- Always
- Only in elastic collisions
- Only in 1D
- When no external impulse acts
Show answer
D: When no external impulse acts. Internal forces cancel.
4. A 3 kg ball moving at (4i + 2j) m/s receives impulse (−6i + 3j) N s. What is its new velocity?
- −2i + 5j
- 2i + 3j
- i + 3j
- 6i − j
Show answer
B: 2i + 3j. v = u + I/m = (4i + 2j) + (−2i + j).
5. A 0.2 kg ball hits a wall at 10 m/s and rebounds at 6 m/s. What is the magnitude of the impulse?
- 0.8 N s
- 16 N s
- 3.2 N s
- 2 N s
Show answer
C: 3.2 N s. 0.2(6 − (−10)) = 3.2.
6. What does the area under a force–time graph represent?
- Impulse
- Work done
- Power
- Acceleration
Show answer
A: Impulse. ∫F dt.
7. Two particles of 2 kg and 3 kg move at 5 m/s and −1 m/s and coalesce. What is their velocity?
- 2 m/s
- 0.4 m/s
- 3 m/s
- 1.4 m/s
Show answer
D: 1.4 m/s. (10 − 3)/5 = 1.4.
8. What are the units of impulse?
- N m
- N s
- kg m s⁻²
- J
Show answer
B: N s. Same as momentum.
9. A gun of mass 4 kg fires a 0.02 kg bullet at 400 m/s. What is the recoil speed?
- 8 m/s
- 0.5 m/s
- 2 m/s
- 20 m/s
Show answer
C: 2 m/s. 0.02 × 400 = 4 × v.
10. In 2D, how is momentum conserved?
- In each direction (component) separately
- Only in the direction of motion
- Only the magnitudes add
- It is not conserved
Show answer
A: In each direction (component) separately. Vector conservation.
Keep going in Full Marks
Full Marks has 51 questions on Momentum and impulse, including in vector form (41 more than here), set out in practice levels. You also get notes and 3D models for all 5 subtopics of Further mechanics, plus games, timed exams and past papers.