A2 June 2024 Q4
4. A particle \(A\) of mass \(2m\) is moving in a straight line with speed \(3u\) on a smooth horizontal plane. Particle \(A\) collides directly with a particle \(B\) of mass \(m\) which is at rest on the plane.
The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e \gt 0\)
After the collision, \(B\) hits a smooth fixed vertical wall which is perpendicular to the direction of motion of \(B\).
The coefficient of restitution between \(B\) and the wall is \(\dfrac{1}{2}\)
Find, in simplified form, in terms of \(m\), \(u\) and \(e\),
| Scheme | Marks | AO |
|---|---|---|
  | ||
| CLM: | M1 | 3.1a |
| \(2m \times 3u = 2mv_A + mv_B\) \((6u = 2v_A + v_B)\) OR \(2m \times 3u = -2mv_A + mv_B\) \((6u = -2v_A + v_B)\) | A1 | 1.1b |
| Impact Law: | M1 | 3.4 |
| \(3ue = -v_A + v_B\) OR \(3ue = v_A + v_B\) | A1 | 1.1b |
| Solve for \(v_B\) | M1 | 2.1 |
| \((v_B =)\ 2u(1 + e)\) * | A1* | 2.2a |
| (6) |
Notes
M1: Use of CLM, all terms required, dimensionally correct (mass \(\times\) velocity in each term). Mass and velocity paired correctly. Condone sign errors on velocities. Condone consistent extra \(g\) and/or consistent missing \(m\) (in every term).
A1: Correct unsimplified equation.
M1: Correct use of Impact Law, dimensionally correct, condone sign errors on velocity.
M0 if separation and approach are on the wrong sides.
A1: Correct unsimplified equation, the direction of \(A\) must be consistent with their CLM.
M1: Use their correctly formed equations to solve for \(v_B\)
A1*: Given answer correctly obtained and exactly as printed. Working should include an equation in \(v_B\) only before reaching the given answer.
| Scheme | Marks | AO |
|---|---|---|
| | ||
| Solve for \(v_A\) | M1 | 3.1a |
| \(v_A = u(2 - e)\) OR \(v_A = u(e - 2)\) | A1 | 1.1b |
| Complete and correct explanation: \(0 \leqslant e \leqslant 1 \;\Rightarrow\; v_A \gt 0\) \(\Rightarrow A\) continues to move towards the wall \(\Rightarrow A\) will collide again with \(B\). OR Complete and correct explanation: \(0 \leqslant e \leqslant 1 \;\Rightarrow\; v_A \lt 0\) \(\Rightarrow A\) continues to move towards the wall \(\Rightarrow A\) will collide again with \(B\). | A1 | 2.4 |
| (3) |
Notes
M1: Use given answer in (a) to solve for \(v_A\). If seen in (a) it must be used in (b) to earn this mark.
A1: Correct expression seen for velocity of \(A\) after impact.
A1*: Correct and complete explanation with no incorrect statements. Must include all of:
- \(0 \lt e \leqslant 1\) or \(0 \leqslant e \leqslant 1\) or \(0 \leqslant e \lt 1\) or ‘for all \(e\)’
- \(v_A \gt 0\) or \(v_A \lt 0\) (must be correct for their \(v_A\))
- Must refer to the wall.
- \(A\) continues to move with unchanged direction or towards the wall (eg do not accept descriptions for direction of travel as ‘to the right’ or similar)
- Conclude second collision between \(A\) and \(B\)
| Scheme | Marks | AO |
|---|---|---|
| Rebound speed or velocity of \(B = \pm\dfrac{1}{2} \times 2u(1 + e)\) | B1 | 3.4 |
| \(\pm m\left[-u(1 + e) - 2u(1 + e)\right]\) | M1 | 3.1a |
| \(3(1 + e)mu\) | A1 | 1.1b |
| (3) |
Notes
B1: Correct unsimplified expression for rebound speed or velocity of \(B\), may be seen on diagram or elsewhere in working. Accept positive or negative.
M1: Attempt to find difference in momenta, dimensionally correct (mass \(\times\) velocity). Must use given answer from (a) for Vb. Condone use of \(e\) or \(e'\) in place of ½
A1: Correct answer, must be positive. Accept any equivalent form that is either factorised or reduced to 2 terms eg \(3(1 + e)mu\), \(3mu + 3emu\)
| Scheme | Marks | AO |
|---|---|---|
| Attempt at KE loss of \(B\) | M1 | 3.4 |
| \(\dfrac{1}{2}m\left[\big(2u(1 + e)\big)^2 - \big(u(1 + e)\big)^2\right]\) | A1 | 3.1a |
| \(\dfrac{3mu^2(1 + e)^2}{2}\) | A1 | 1.1b |
| (3) | ||
| (15 marks) |
Notes
M1: Attempt at KE loss of \(B\): clear attempt at a difference in KE of \(B\) before and after impact with the wall. Must use velocity of \(B\) from (a). Condone use of \(e\) or \(e'\) in place of ½. Dimensionally correct, condone subtraction either way round.
A1: Correct unsimplified expression for KE loss of \(B\)
A1: Any equivalent factorised form eg \(\dfrac{3mu^2(1 + e)^2}{2}\), \(\dfrac{3mu^2(1 + 2e + e^2)}{2}\), oe