A2 June 2025 Q5
5.
Three small balls, \(A\), \(B\) and \(C\), have masses \(2m\), \(3m\) and \(4m\) respectively.
The balls are initially at rest in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\), as shown in Figure 2.
Ball \(A\) is projected towards \(B\) with speed \(5u\) and \(A\) and \(B\) collide directly.
Immediately after the collision, the speed of \(B\) is \(3u\)
The balls are modelled as uniform spheres with equal radii.
After the collision between \(A\) and \(B\), ball \(C\) is projected towards \(B\) with speed \(u\)
Balls \(B\) and \(C\) collide directly.
The coefficient of restitution between \(B\) and \(C\) is \(f\)
Given that there is a second direct collision between \(A\) and \(B\)
| Scheme | Marks | AO |
|---|---|---|
  | ||
| Use of CLM: | M1 | 3.4 |
| \(2m(5u) = 2mv + 3m(3u)\) OR \(2m(5u) = 2m(-v) + 3m(3u)\) | A1 | 1.1b |
| \(v = \dfrac{1}{2}u\) OR \(v = -\dfrac{1}{2}u \;\Rightarrow\; |v| = \dfrac{1}{2}u\) | A1 | 1.1b |
| (3) |
Notes
M1: Use of CLM for \(A\) and \(B\), to form an equation in \(v\) and \(u\) (and \(m\)). Dimensionally correct with correct mass and velocity pairings. Condone sign errors.
May use two impulse-momentum equations: \(-2m(v - 5u) = 3m(3u - 0)\)
A1: Correct unsimplified equation.
A1: Correct only, must be positive.
| Scheme | Marks | AO |
|---|---|---|
| | ||
| Use of NEL | M1 | 3.4 |
| \(3u - v = e \times 5u\) OR \(3u + v = e \times 5u\) | A1 | 1.1b |
| \(\Rightarrow e = \dfrac{1}{2}\) | A1 | 1.1b |
| (3) |
Notes
M1: Correct use of Impact Law for \(A\) and \(B\) to form an equation in \(u\) (and \(v\)). Allow consistent omission of \(u\). Dimensionally correct, condone sign errors on the velocities.
M0 if separation and approach are on the wrong sides.
A1: Correct unsimplified equation.
A1: Correct only.
| Scheme | Marks | AO |
|---|---|---|
  | ||
| Use of CLM | M1 | 3.4 |
| \(3m(3u) + 4m(-u) = 3mx + 4my\) OR \(3m(3u) + 4m(-u) = 3m(-x) + 4my\) | A1 | 1.1b |
| Use of NEL | M1 | 3.4 |
| \(y - x = f \times (3u - -u)\) OR \(y + x = f \times (3u - -u)\) | A1 | 1.1b |
| E.g. \(\begin{cases} 3x + 4y = 5u \\ y - x = 4uf \end{cases} \;\Rightarrow\; x = \dfrac{u}{7}(5 - 16f)\) o.e. OR E.g. \(\begin{cases} -3x + 4y = 5u \\ y + x = 4uf \end{cases} \;\Rightarrow\; x = \dfrac{u}{7}(16f - 5)\) o.e. | M1 | 1.1b |
| \(v \gt x \;\Rightarrow\; \dfrac{1}{2}u \gt \dfrac{u}{7}(5 - 16f)\) OR \(v \gt -x \;\Rightarrow\; \dfrac{1}{2}u \gt -\dfrac{u}{7}(16f - 5)\) | dM1 | 3.1b |
| \(\dfrac{3}{32} \lt f \leqslant 1\) | A1 | 2.2a |
| (7) | ||
| (13 marks) |
Notes
M1: Use CLM, dimensionally correct, with correct mass and velocity pairings. Must have all non-zero velocities. Condone sign errors. Follow their directions for the unknown velocities of \(B\) and \(C\) after impact. (Ignore the diagram if it benefits the candidate.)
A1: Correct unsimplified equation.
M1: Correct use of NEL for \(B\) and \(C\), dimensionally correct. Condone sign errors on velocity. Must have all non-zero velocities. Condone use of another letter for \(f\).
M0 if separation and approach are on the wrong sides.
A1: Correct unsimplified equation, the directions of \(B\) and \(C\) after impact must be consistent with their CLM equation.
M1: Solve simultaneous equations with two unknowns to find an expression for the velocity of \(B\) after the second collision in terms of \(f\) and \(u\). Condone use of another letter for \(f\).
Must use CLM and NEL to reach \(x = \ldots\) or a multiple of \(x = \ldots\)
dM1: Dependent on 3 previous M’s. Complete method to find the values of \(f\) for a second collision. Must consider \(A\) and \(B\) moving in the same direction and the speed of \(A\) > speed of \(B\).
A1: \(0.094 \lt f \leqslant 1\) (0.094 or better) Both ends required with correct inequality signs and must use \(f\) as coefficient of restitution.